In order to 100% complete a district there are some number of the following things that have to be completed:
Black Cat Stakeouts
Not all districts have everything or the same amount of each item. Once you have 100% completed all districts you will earn this trophy.
Each item will automatically appear on the map after completing a Main Mission or Side Mission. The last two things will be available shortly after starting Act III. Taskmaster Challenges only have to be completed, no specific score is necessary. Crimes and bases only have to be completed as well. You are not required to get all bonus objectives.TheOdd101
mentioned a tip to help spawn crimes, but I had no problem getting them to spawn by just setting a waypoint at the other end of the district and just swinging to it, regularly pressing in the right stick along the way.