For this achievement you simply need to turn one district hostile of the four. Pillars of the community bring down the meter the fastest, although killing off lower civilians works just the same. In order for the district to fully be considered hostile though you need to get the newspaper article on the fall of that particular district. You can simply rest enough days in a row for the people to die of their ailments--that works just fine.
I suggest pairing this with collecting all of the weapons / collectibles so that you can clear out a district, kill off civilians, and then turn it hostile by resting in the end. This also works great when going for: