Given the nature of the game, there may be other ways of completing this achievement. However, this should be the most straight forward method.
Once you finish downloading The Pitt and start up a saved gamed, shortly after you get in game you should receive a message about new radio stations. Open your pip boy and listen to Werhner's station. This should give you a quest to meet him in the far north-north-west section of the map, almost off of it. If you have this area explored, fast travel to the nearest location. If you don't, prepare for a very long hike.
When you arrive there and speak with him, he'll ask you for help to free his people but will first tell you that you need a slave outfit so you can hope to fit in. He will then send you to the tunnel which happens to have a slave camp in front of it.
When you arrive there, you'll find 3 slavers guarding 3 slaves. You'll need access to the slaves. There are 2 ways to go about this, pay for them with 600 caps or kill the slavers and get the key off of one of the bodies. Either way, when you open the cage, one of the slaves will talk to you. You'll need to ask him about one of the slave outfits. He'll tell you to grab the one of the dead body, if you do so, you get Good karma points (My guess is that if you demand his instead, you get Bad point, same if you kill him). Enter the tunnel then use the hand cart to travel to the Pitt.
When you arrive, there will be a fire fight. Afterward, travel forward until you run into a gate that is locked. Wait for Werhner to unlock it. While here, he will tell you they will take away all your items when you enter the Pitt but you can convince him to hid a knife (ouch) or pistol (can't even imagine the pain) on you.
After this point, travel forward until you reach the bridge. It is best to avoid the bombed out building on this side, there are only 3 enemies in it and nothing worth looting. Travel across the bride. Watch your feat while on it, there are several mines in the area. While traveling across, you may have to fight some stray dogs (unless you have the animal magnetism perk) and a sniper at the opposite end.
After making it across, you'll see some slaves running towards you who will inevitably run into the mines. Shortly after this, you'll run into the front gate of the Pitt. On the other side of it, you'll have a conversation with a guard who will proceed to take your items. Once this conversation is over, this achievement will unlock.