PAY ATTENTION: This guide was written a long time ago and is now obsolete. There were lots of changes made to the game since then, and in different aspects, so I can no longer guarantee that the recommendations in this guide are helpful or even applicable now.
Maybe one day I'll update this guide for the current game version, but for now read this at your own risk.

Some achievements are hard to get because they're very luck based (e.g., "Outside Context"). Other achievements just need lots of work to do (e.g., "Very Open Borders"). This particular one requires both luck and hard efforts, because you have to both be lucky to get the "War In Heaven" event AND to be more than prepared to become the leader of the "League of Non-Aligned Empires" and win the war.
This guide is designed to help you to do everything possible to increase your chances of success. So, let's get started.

NB: If you don't feel like reading the whole guide (which is understandable, it's quite humongous) — you can scroll down to the "Conclusion" part where all the key parts are listed briefly.

Comment: In order to increase your chance to trigger the WiH event, you must make sure that all 4 biological Fallen Empires spawn in galaxy.
Comment: Max amount of AI empires is 24 / 30, but you don't want to set that, because there was a bug (IDK, maybe it's fixed already) that actually prevented spawning of one of the Fallen Empires due to lack of place in galaxy. But still, you need the big amount of AI empires, because you have to make sure that enough of them will make it till the Endgame and you'll actually have someone to form the "League of Non-Aligned Empires" with (and that you'll have enough allies that won't die too soon and dismantle the League leaving you as its only member),
Comment: The faster you'll be prepared till the Endgame, the better off you'll be.
Comment: You can set more, it's up to you. I prefer 1x, because I rely on Habitats rather than planets.
Comment: The more there will be, the more of them you'll be able to enlighten and get free monthly 0.25 inluence for each.
Comment: Crisis spawning in the middle of WiH is not the thing you'd like to have.
Comment: This is the year, from which Fallen Empires start to awaken. You don't want to set it too far, because, first, you don't want to spend way too much time waiting for it (since you most likely won't get the event on the first try), and, second, you also don't want all the other AI Empires to get too strong and steal the leadership in the League from you due to having >100k fleet. But setting the year too soon is not good either, because you just risk being unprepared till the start of the war. From my experience, 2350 is probably the golden mean here. But, if you are still afraid of risk, you can move it 25 years forward, to 2375.
Comment: For some reason these difficulty settings affect only regular AI Empires, but not the precursor ones. Set it too high — and you can kiss the Fallen Empires goodbye, AIs will just crush them beforehand.
Comment: It would be best to ensure that all AI Empires get evenly developed throughout the game.
Comment: I make it this way because it gets much easier to get connected to each neighbor country only by one or two hyperlanes, and that helps to wage war with them (more about that later) and not let their forces in your borders. This is not the best choise, its disadvantage is making the way to each of the Awakened Empires longer and harder, so feel free to change it if you want.
Comment: As you know, you are only allowed to use gateways if both the departure one and the destination one are in either your borders or the friendly empire's borders. Fallen and Awakened Empires, however, are able to use all the gateways, no matter where are they located. That's why it would be best to have them nowhere.
Comment: Wormholes are pretty messy, they allow free movement for everyone and connect to each other in irrational way making pathfinding for both your and AI's fleets pretty much unpredictable. Of course, there's a good thing about them — they allow using sensors and keep vision on what's happening on the other side. But if you build Sentry Array, you'll have that vision anyway.
Comment: Pretty obvious, no Ironman — no achievement.

There is no perfect setup here, many different variants are possible. However, some things are worth mentioning.
1. You have 150 years to build a powerful empire with a strong fleet. That means wars will be inevitable, either for conquering planets, getting access to tasty systems or just making subjects.
2. Despite that, your relationship with all the AI empires actually matter. It would be best to have open borders and positive attitude (opinion) with most of empires. Fallen Empires count here too — you DON'T want to anger any of them before the WiH. Angry FE declaring you war when you're already in the middle of WiH won't help you at all.
3. You will also need a big resource income to both build and maintain the large fleet and also to develop your colonies and megastructures.
That means, Pacifism and Fanatic Xenophobe ethics are not an option. Militarism, on the other hand, is very useful. And so is Spiritualism, since it makes going for Psionic ascension path a bit easier (more about that below).
As for the Ascension path, I recommend going for Psionics. You see, your foremost priority should be the fleet power and technology, and Shroud can give you everything for that — yes, I mean Psionic Shields (which are the best in game), Precognitive Interface and Psi-Jump Drive. Also, if you're lucky enough, it can boost both your fleet efficiency (whether by fire rate, shield power or upkeep) and economics (by increasing your income). Bio ascension path can empower your pops and your army, but does almost nothing to your fleet. And Synthetic ascension path is just out of question, since it angers Spiritualist Fallen Empire, making them your sworn enemy forever.

Here's what I have used for my empire:
Ethics: Fanatic Authoritarian (free +1 influence) and Militarist (+10% fire rate)
Civics (at the start): Mining Guilds (+10% minerals production), Environmentalist (-15% consumer goods cost)
Civics (after reformation): Mining Guilds (+10% minerals production), Feudal Empire (-50% to the "Relative power of empires" modifier; this one will be explained later), Distinguished Admiralty (+2 admiral level cap, +2 leader capacity)
Ascension perks: Technological Ascendancy (+10% research speed), Interstellar Dominion (-20% to starbase and claim influence cost), Mind Over Matter (psi path, 1st stage), Transcendance (psi path, 2nd stage), Voidborne (habitats), Master Builders (increased habitat capacity, megastructure build speed and also enables the Mega-Engineering technology), Galactic Wonders (megastructures), Galactic Contender (increased damage to Fallen and Awakened empires)

Of course, if you don't have the Utopia DLC, your perks will surely vary. In that case, setting more habitable worlds in galaxy can be a really viable option for you, and choise of perks then can be shifted to something that helps planerary management (Mastery of Nature will work fine, for example).

So, you start at 2200 and have 150 years to make a powerful empire with a strong fleet and heavy army. In order to achieve that, you need to advance your technologies, expand, and gather enough resources.
The strategy is to stay in one system for some time and rely on just surveying the neighbor systems with multiple science ships. This way the Discovery tradition tree can be utilized to max efficiency. And, surely, the edict "Map The Stars" will be the really big deal here. In this "peaceful" period it would be best to pick the technologies that improve your buildings, empire political mechanisms or resource gathering rather than ship tech; and resources can be spent into upgrading the existing buildings and constructing the new ones. Early active surveying means finding lots of uninhabited systems which can result in anomalies leading to interesting Special Projects — precursor empire research, finding free ships, alien box with mixtures that can improve your race, or even the Gaia World with 25 tiles (which can be used for a new capital).
The second tradition tree to complete is Expansion. Completing it makes expanding cheaper and faster — and that's when you start to take new systems and found new colonies.
It's worth mentioning, though, that expanding should best be made relatively to how hyperlanes go and where are the other AI empires' borders. Best case scenario will be locking them all down, leaving the unclaimed space for only yourself.
Picking the next traditions mostly depend on how you prefer to play. I've gone with Domination tree next. Why? Well, the strategy I used is all about making subjects. Tributaries, in particular. See, the AI got really dumb and that's why turning the other empires into your taxed subjects rather than conquering them is basically killing two birds with one stone — you don't have to hope they will help you in war, like vassals, and you don't have to manage additional planets, like if you just took their planets away — you just get free profit and manage your own worlds in the meantime. Besides, unlike vassals, tributaries don't participate in your wars at all, which means that turning all your neighbors into tributaries also makes them work as neutral buffers against the future enemies that lie further.
Oh, and by the way, here comes the "Feudal Society" civic I mentioned earlier — helps much, when you get way too many of those subjects.
So, since you don't have to bother yourself with conquered worlds, you can spend time into developing Habitats and Megastructures. Taking all 3 Ascencion perks connected to them helps much, because that gives you both increased space in Habitats and increased build speed, as well as the Mega-Engineering tech, so you don't have to wait for it to pop up in its own, which can take centuries (I'm not kidding).
The best megastructure to start with will be Sentry Array — doesn't take much resources, and musthave for knowing where all the fleets go in galaxy. Remember, not only you should know where to strike first during the WiH itself, but also you have to make sure that nobody in galaxy, except you and Fallen Empires, has fleet worth of 100k before the endgame.
If you finished it and still have time for something else — go for Dyson Sphere. Energy income will be as much needed as mineral, and additional energy can be used for Resource Replicators to boost your minerals.
Keep an eye for your Starbase capacity. If you see you can make more — make more and fill them with Anchorages, naval capacity will never be enough.
And, obviously, keep making fleets and armies! Armies should be produced constantly on at least one planet (would be better to have a Military Academy there), and it would be best to not have just regular assault armies, but some powerful ones — whether those are Psi forces, Gene-mods, Androids, Xenomorps or smth else. Fleets, on the other hand, would best be built in a balanced way — enough to win subjugation wars without losses, but not way too many to save up your mineral and energy income. I've gone for full kinetic battleship composition, but I won't say it's an imba — use whatever you like most. What I can say for sure is that your fleet must always be technologically advanced, because delayed upgrades will take very, very long time.
Speaking of technologies, not only will the Shroud gifts help. You can also get something from Leviathans. For example, Dimensional Horror gives you Jump Drive research tech, Ether Drake can give you access to the Dragon's Hoard planet, after claiming which you can get the level VI armor (best in game) and Enigmatic Fortress can give Enigmatic Encoder & Enigmatic Decoder.
Also you can try and get something unique from one of the Fallen Empires. But that's risky. Remember, you don't want to anger them much. So, no border friction for Xenophobes, no Holy World desecration for Spiritualists, no purges and livestocking for Xenophiles, no insults, no rivalries. Just claim one system, conquer it, finish with Status Quo — that's it. Best case scenario will be killing not all their fleets (so they can recover till the endgame), but just enough to get the debris you're looking for. Another way is to start the war, kill some of their ships (so there will be debris) and surrender immediately. Humiliation is not the best thing to have, but still it won't be much of a deal.
Regardless, your goal is to get ready till the endgame, which means having:
1. Fleets enough to crush both Awakened Empires. 1000 units of naval capacity is fine, 1200 is even better.
2. Armies enough to conquer all the planets fast enough and without needs of reinforcement. 24k of army power worked for me.
3. Resources enough to provide upkeep for all of this. You can test it out by taking your fleets away from starports — if energy and minerals severely decrease and don't have enough in storages, then you're doing something wrong.
4. Intel on most of the galaxy, to see where the Awakened Empires will go and what they will do.
5. AI empires having less than 100k of total fleet power. If the endgame is at 2350 and difficulty level is low, it surely won't happen, but if you see someone actually managed to make such a fleet — cut it down by all means.
6. Fairly good relationships will all AI empires, whether they're independent, subjects or Fallen.
7. No war. Peace. You risk not getting a question about leading the League, if you're at war.

If you have done all your best and increased your chances for successful achievement, all you need to do is just wait. Wait for the Fallen Empire to Awaken, then wait for it to mark another Fallen Empire as its "ancient rival", and then wait for that "bitten" Fallen Empire to awaken as well. It all can take a decade, so be patient.
But, of course, this event is not guaranteed, so if you've waited for 25 years and no rivalry happened, then you have nothing to do but to drop it and start all over.
And if that Awakened Empire has thrown the rivalry and started the war immediately without even waiting for its rival to awaken, don't panic. It happened to me once. I've seen the Xenophile Awakened Empire declaring rivalry on Spitirualist Fallen Empire, starting the war and turning them into signatory, but soon after that those Xenophiles have marked Fallen Xenophobes as their new rival, and then has happened the real WiH, with awakening, gathering subjects, forming the League and stuff.

Anyway, after you've seen who's your future enemy, it would be wise to move your fleets and armies directly to the borders of one of them. It won't be long before the war starts, and you'll receive two questions — the first one about joining one of the sides (where you decline both) and the second one where you're offered to lead the League of Non-Aligned Empires (and this one you accept). Wait for a couple of months now, all the other still independent empires will be asked about joining your League, and you'll have to accept them one by one. But don't wait too much, gather a decent alliance and start the war immediately against both Awakened Empires.
You have to work quickly. See, the achievement requires you to lead this League to victory, and since it works just like a regular federation, your leadership will be lost in 10 years, so this is the time for you to crush both of the Awakened Empires.
It isn't so hard as it sounds afterall. The good news is that your actual enemies will be only the Awakened Empires themselves, while their subjects will appear neutral to you. So the victory contidion is to conquer all their planets and also all the new planets they've managed to take from the others. Keep your fleets together at first to crush their powers, later try splitting, if there will be need to catch and neutralize those that slipped away. Keep your armies alive, occupy the planets one by one, and victory will be in your grasp.

To conclude everything written above, I'll highlight the key points of this guide.

• Make such a galaxy, that can contain all 4 biological Fallen Empires.
  
• Set the endgame year to 2350, so you will get prepared till the endgame, while other AI empires won't.
  
• Expand your empire, develop your economy and build fleets & armies. You'd best make 1000—1200 units of ships, >20k of army power and have enough resources to not have problems with upkeep.
  
• Utilize habitats and megastructures to your advantage. Sentry Array, in particular, will help you a lot.
  
• Make sure that no AI empire gets >100k fleet power.
  
• DO NOT anger the Fallen Empires beforehand.
  
• For this reason, DO NOT pick the Synthetic ascension path. Psionic ascension path, on the other hand, is recommended.
  
• Keep an eye for technological advancement of your fleet — aside of making research, look for the new tech in Shroud and try getting some from Leviathans.
  
• You should better be at peace (or at least on the way to it) when the endgame starts.
  
• After the start of the endgame, wait for some Fallen Empire to awaken, declare rivalry to some other Fallen Empire and start the War in Heaven (not just regular war). If 20—25 years pass and no rivarly happens, then drop it and start all over.
  
• After you have become the leader of the League and gathered enough allies, declare war on both Awakened Empires and do whatever it takes to eliminate them both in 10 years, so you won't lose your leadership.

Good luck & have fun! :)

Thanks for people who helped me to adjust and correct the guide:

• Viera — for confirming that you actually must be the current leader of the League in order to get the achievement.