Streets of Rogue

Each preset character in the game has a set Big Quest, which is essentially the same side mission in every level. To unlock this achievement, you must complete that side mission in every single level from start to finish of a playthrough. For example, the Soldier's Big Quest is to destroy each generator in every level. There are anywhere from one to four generators per level, and you need to find them and destroy them (just shoot them until they explode). The Vampire, on the other hand, is tasked with finding and killing the werewolves in each level. Each character has their own, and the task is the same in every level, it's just a question of how many enemies, objects, etc., per level. Check with  in the upper right corner routinely to ensure it's marked green when you complete it on that floor. You'll also get a significant XP boost for completing it on that floor. Stick with it for an entire playthrough, and when you finish the third level of Uptown, you'll unlock this achievement if you've done them on every floor.

As a side note, remember you can utilize the save exploit discussed in the roadmap in case anything goes really wrong while trying to complete these side missions.

I personally did this with the Soldier on my first run, and had no real issues. I used the machine gun and Infinite Ammo mutator and shot my way into every building, then shot the generators enough to let them explode a few seconds later after I'd escaped. If I accidentally screwed up and took a ton of damage while trying to do this, I'd use the save exploit to try the level again.