Murphy is a Lieutenant in the Chicago Police Department, in charge of the Special Investigations (or "S.I.") division, which is appointed to all cases in which something unexplainable or supernatural is involved. In the past, transfer to SI has been a way for the department to get rid of out-of-favor officers. Murphy, on the other hand, took the post seriously, going so far as to hire the only wizard in the phone book, Harry Dresden.
Warning: The following contains spoilers for later books. Read at your own risk.
Relationship with Harry
The events of Storm Front and Fool Moon cause Murphy to distrust Harry, and she reacts in an outwardly hostile and violent manner when she suspects he had a hand in the victims' deaths. She attempts to arrest him in both books. However, the events of Grave Peril and Summer Knight lead her to put more trust in him, and she becomes one of his closest friends.
She admits to some romantic attraction to Harry as well, but is unwilling to act on it for several reasons. Firstly, she worries about complicating or ruining their friendship and working relationship. She also worries about their difference in lifespans: as a wizard, Harry will live for centuries, and she wants someone who can grow old with her and raise a family. Then, there is the reason that Thomas points out: she and Harry are both stiff-necked idiots.
Karrin is the oldest child of Collin and Marion Murphy, and she is somewhere in the range of 27 to 29 years old at the time of Storm Front. She has at least two brothers and at least one sister, whom we meet in Blood Rites. In Blood Rites, Harry learns that before his suicide when Karrin was 11, Collin Murphy was a senior detective in the Chicago PD and a "black cat" investigator, covering the same territory as the modern-day S.I. department.
Murphy has been married and divorced twice, the first time at age 17 to Gregory Taggart, who is some eleven or twelve years her senior. He leaves her, and all subsequent attempts at conversation end in a fight. She later marries Richard Boughton. They also divorce, when Murphy translates Rick's desire to start a family as an effort to keep her at home, away from her job. Some time between Blood Rites and Proven Guilty, Richard gets remarried to Murphy's youngest sibling, Lisa, who is about 11 years her junior. This can only be described as "icky."
In Proven Guilty, Murphy is demoted to Sergeant and replaced by co-worker Detective Sergeant John Stallings.