So I tried the top rated solution and it just wasn't working. I just couldn't keep my tech up high enough while fighting one or two (or even three) wars at once. And the Japanese didn't seem to really have any advantage that was helping. So I tried something else.
Stick to the Lightning Round scenario, but play as the Greeks. Starting with Democracy is hugely beneficial, as you'll want that +50% to science and gold ASAP. Switch to it first turn.
Now, what I found to be the most important bit. The map. Judging by the comments, this is what makes or breaks your game in this case. Restart until you find a map where all three of your cities are, if not on their own island(s), on a remote peninsula with lots of surrounding water and no nearby civs. I restarted at least a half-dozen times in a row until I got a suitable location tucked up near the north pole. When you find a good map, set your government to Democracy and keep a save, just in case things go all caddywumpus later.
The reasoning behind this strategy is you want to avoid other civs as much as possible. More distance between you and the other civs, the better. Less contact with other civs, less war. Less war, less wasting time and resources building your military. I didn't build a single unit all game.
The nice part about playing as the Greeks is that you have a very useful bargaining chip right from the start: Democracy. In both of my attempts, the Russians threatened war if I didn't teach them Democracy. Both times I agreed to their deal, and they promptly switched to Democracy and never bothered me again. While you're giving away a tech and the +50% science, the fact that you keep another civ from declaring war is worth it. The computer is extremely aggressive on Deity, but Democracy keeps them from attacking you. And if they do, they'll have to change governments first which gives you some warning.
In my experience, Democracy is the most common demand you'll receive in the first 70% or so of the game. Later on, you'll get asked for other techs or for money. Keep up to speed on who's winning. Personally, I checked after every time I chose a new tech to research. If you're 5 techs ahead of a civ and they offer 6 turns of peace in exchange for a low level tech, let them have it. As much as I hate appeasement, it's often the best option on Deity, especially when you don't have much of a military to speak of. However, try not to give away techs to civs that have more than you. Which brings me to another reason that being isolated is nice.
Distance from other civs is useful because it discourages war. I had the French randomly declare war fairly late game from clear across the map. Several turns later, after seeing no French troops, ol' Nap asks for a treaty. The computer just isn't willing or capable to send troops very far if there's an ocean in the way, and will often just give up.
As far as actual strategy goes, put everything into getting libraries built right away. Full production, save for one food square per city. Once those are built, focus mostly on science. Build a courthouse in cities where there's a lot of nearby water if you really want to take advantage or granaries if you want to boost population faster, but every other building just isn't worth it. Neglect your culture and let your people become Disgusting, there's no one around to culture-flip your cities. Great people aren't super helpful in this scenario. But as the Greeks, you get more of them which is nice.
One important thing regarding Great Inventors in particular. You'll almost always want to settle them in a city, giving it a 50% boost to science. However, using them to immediately complete a tech can win you the game. I passed up on completing Gunpowder (needed 3 more turns) and ended up losing the game 2 turns later. I was at 23/24, and could have won the game right then if I used the Great Inventor better. So keep tabs on how many techs you have left!
Build a 4th city if you have a good place for one by the ocean (if you get free settlers, that's a good time to do so) and just keep increasing the population of your cities while keeping the science up. I was getting new discoveries every 2 - 4 turns throughout the game. Always pick the fastest one to discover. Democracy is the only one that would be any help, and you have it already.
Hope this is helpful, I got this achievement on the second try using this method. On my first try I forgot to switch to Democracy until roughly halfway through the game, which cost me quite a bit of science.