The Earliest you can do this achievement is towards the end of Sequence 5, so I suggest saving as much money as possible upto this if you want this quickly.
Every district has a Borgia Tower. These need to be destoryed and taken over as part of another achievement.
Once the tower is destoryed, you can start renovating districts. This is simple, destroy the Antico district tower and build renovations. The quickest ones are shops, Blacksmith, Art and Stables are cheap. There are also some historical locations and Aqueducts you can renovate in this district.