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After you've found Arthur's bounty hunter in the depths of Gimble's Prosthetics during the A Bounty For the Hunter
quest, he will refuse to return to work for Arthur. Therefore, when you tell Kilpatrick this news, he will ask you to track down a man named Mike Durbin or, more affectionately, "Muddy Mike". Since you have no idea where Mr. Durbin is, your first visit will need to be the apartment directly across from yours above Trip's Pawnshop, where Marian Murietta, the girl who put up money for Mike's bail bond, lives.
Leave Kilpatrick's and return to your Haven. Feel free to check your email for any correspondence you might have (possibly opening other side quests), then search behind the plant in the hallway outside your door. You'll find a Spare Key, which will open the door to the apartment across from yours (although you can certainly pick the lock instead, if your skill is high enough). Inside this apartment are two items of interest. First, there's a Peepin' - A Voyeur Field Guide
manual near the oven. Secondly, you can listen to the answering machine to hear a message for Marian from Mike stating that he is heading downtown to the residence of a Mr. Milton in the Skyline Apartments, room 2A. Listening to this message will get you one point of Experience.
Run back to Kilpatrick's and tell him what you've learned about Mike's whereabouts for another point of Experience and $150 in cash. You're not quite done yet, though. The completion of this quest triggers another one called Mudd Hunt
, which will require you to visit both the Downtown and Hollywood districts.