Beat all levels without dying. You can track it by viewing the four trophies under the level name in the level select menu - the third trophy is the one for beating the level without dying. Solo or co-op doesn't matter, but it seems like no one gets the trophy if anyone died in co-op.
An easier way to do it could be to play local co-op with multiple controllers, just to give yourself more abilities. Using The Mole and The Lookout, for example, could be a great combination. The best, in my opinion, would be The Mole, The Lookout, The Pickpocket and either The Cleaner or The Redhead to deal with those lone guards in pesky positions. You could leave The Lookout in the entrance where he won't be found, but his ability to show enemies from further away stays active, and you can use The Mole to open some choice paths and then come back with The Pickpocket to grab gold without being seen.
(If you're having trouble seeing the trophies in the menu, like I initially did, turn up the game's brightness to like 80% (it starts on 20%).)