To unlock the The Cat Came Back trophy in Spider-Man The City That Never Sleeps: The Heist, you simply need to finish the first mission of the DLC. You can access that DLC, assuming that you have purchased it, through the main game's menu.
Note that the DLC takes place in its own instance of New York, the missions aren't integrated into your main save. In fact the story takes place after the main campaign so it's strongly recommended that you finish at least the story missions of the main game first.
As for getting this trophy, the first mission of the game is not particularly hard. You will mainly be facing off against Fisk-like thugs though they run under a different name. There is one slightly tricky section where you have to stop specific bad guys from making off with some stolen goods while being harassed by many other bad guys, but all in all it's not too difficult. This will unlock at the conclusion of the first story mission.