You lose reputation for various tasks.
Denying a demand by a faction, attacking a faction with high reputation, stepping out of an alliance or other diplomatic option like it, destroying npc settlements, and razing towns.
There are other factors, but this was my strategy.
I played a 3 player scenario.
I got into an alliance with one player, got into integration with an NPC, and built 5 cities. It takes 3 turns to raze a city, so I put all 5 cities to raze turn one, left my alliance turn 2, then went to war with both the npc faction (spacers for me) and my previously allied player turn 3.
This shot my reputation down to -600 and landed me the achievement immediately upon my next turn, as the 5 cities lay in ruin.
Happy hunting (>*_*)>