Wwau's solution pretty much explains what you need to do for this achievement. However, if you're finding it difficult to complete some of the activities, I was able to replicate a glitch from the original Saints Row The Third (Xbox 360) game to pretty much net you this achievement with a bit of saving and replaying.
"Unlimited City Takeover" glitch - step-by-step:1. When you reach the mission "Stag Party" in Act 2, make a manual save before it.
2. Proceed with the mission - go to Stag HQ to burn the banners, return to Saints HQ to defend your turf.
3. At the end of the mission,
choose to give Josh to Cyrus in exchange for the City Takeover bonus. Parachute down the building to end the mission.
4. Go through all the end scenes and reward screens until you resume control. Notice on the top left corner, the game will autosave.
5.
DO NOT GO TO YOUR PHONE NOW. Instead, load your manual save that was before the mission.
6. Once the save is loaded, press
to go to your phone. You'll notice now that you have the City Takeover Bonus from the "Stag Party" mission, even though you technically haven't done the mission yet. Use the bonus on any hood - this bonus completes every activity and gang operation, as well as buy every property and store in the hood for you. Collectibles are not automatically collected though.
7. Once you complete the step to take over a hood, make another manual save. Your hood progress is now saved.
8. You can now replay "Stag Party", and redo steps 1-5 repeatedly until you take over all the hoods, without actually having to do the activities. The mission takes around 15 minutes to complete.
All the activities and district control achievements will unlock when you do this.
(You can also use this method to farm tank kills, headshot kills, etc. by making a manual save during the mid-mission. It doesn't save the mission progress, but challenges progress will be saved.)
Now, obviously this is repetitive and boring, and most of the fun in this game are from the activities. I personally used this for activities that I was unable to complete (Heli Assault Hard and Trailblazing Hard). So this is just a trick for those who have tried, but unable to finish the harder activities. Don't flame me!