Start a new game with the "Knights of the Toxic God" origin.
This origin will unlock several new buildings and unit types, as well as an additional starting habitat. As far as I can tell, none of this new content is required to complete this achievement. Looking at the game post-achievement, I don't think I built any DLC specific units/structures.
Getting the achievement itself is pretty straightforward. You have a progress bar at the top right of the screen (similar to a Victory Parade after defeating a Leviathan or when a shortage crisis is brewing). This bar shows progress towards each stage of the event, as well as a final completion time. The completion speed is determined by your total Energy & Unity output (and you can choose to give more or less per month).
There is a series of popups that will give you the story behind the origin and the quest. These will also require you to make choices. I'm not sure that you need to pick a specific combination to get the achievement, but note that YOU MUST NOT CHOOSE THE "END QUEST" OPTIONS THAT POP UP. There will be at least two times when you have the option to totally end the quest, and this will definitely void the achievement.
Things to note:
- Psionic Theory/Mind Over Matter/Transcendence helped me with the quest. No spoilers, but it did.
- There are some options that are meant to trick you. Some NPCs are helpful, some are not. You can guess, but I think at least one might cancel the achievement (Psionics might help... hint hint)
- When you reach the "final boss"... its not the final boss. This is a bit of a spoiler, but if I had to play a whole Stellaris game and picked the wrong option I'd be mad.
The achievement won't pop up immediately after the correct choice at the end. There was a popup about 20 years later ingame, and this gave me the achievement.