To unlock The City is My Family trophy in Spider-Man's Turf Wars DLC, you need to reach 100% completion on the DLC. That includes:
-- All main missions
-- All Screwball challenges (note: You don't need to get Ultimate, but you'll need Spectacular or better for another trophy)
-- The additional Side Mission that appears after you complete all Screwball challenges
-- 5 Hammerhead crimes in Harlem, Upper East Side, Upper West Side and Hell's Kitchen
As soon as you finish all of the above the trophy will unlock and you'll also get the sweet Spider-Clan Mangaverse suit.