A Long Way To Fall
This is a not a time-sensitive mission, except that after you pull a gun on the initial character to get info, if you decide to kill him, you should then run on into the building before his friends start in on you. Even as you wait for the elevator inside, a Bad Guy ("BG") may follow you in, so have your combat shotgun (or other weapon) ready.
As soon as you can after getting off the elevator, open fire on the two guys there, then go through to the doorway on the left, watching for someone in the open doorway in the hall. After dealing with him, circle the floor twice to make sure you get all the BGs. You should not have to worry about any coming up behind you or from behind closed doors. Just be prepared to shoot as you round each corner.
After clearing the floor, back up the stairs aiming above you and clear the next two floors. When you get to the floor with the yellow marker in front of a door to your right, go to your left and clear the floor, then enter the apartment where three BGs are waiting.
There is a first aid kit in the kitchen if you need it. The boss will eventually escape up to the roof. Follow him up the stairs, aiming above you as before. When you come out on the roof, you can see the red marker above the boss, but not the boss himself. A BG or two may show up. After you dispense with him/them, just walk over to where you saw the red marker and finish off the boss.
Pretty easy, but due to the large number of BGs, you will want to start with full health and armor.