Murphy must have discovered more than we were told about in the first couple of books. Remember at the beginning of Fool Moon when she came in to talk to Harry at Mac's just after Kim had left the table. Murphy picked up the crumpled note that contained Kim's drawing of the summoning circle and placed it in her coat pocket.
Now we could surmise that while Murphy had been checking out the werewolf murder scene she had Carmichael tail Harry, so that is how she knew where he was, except Harry wasn't even a little surprised to see her there. This means Murphy had been to Mac's before; probably with Harry, but we didn't see her there in Storm Front, did we? So perhaps sometime Harry invites Murphy to have a beer. It could have happened before Storm Front. By the start of Fool Moon Murphy was casually gathering information on Harry when she picked up the drawing. She didn't have a specific reason to do so, but she held onto that drawing. I wouldn't be surprised if part of Murphy's backstory is going to McAnally's when Harry wasn't there and talking to Mac or some of his patrons to see if she could learn anything, either about Harry or the supernatural world in general.
I doubt Murphy would have learned very much, but she could have heard about the White Council or at least about a governing body for people like Harry. The things Murphy might have picked up about Harry could have been both reassuring and disturbing. All it would have taken is one person to say something like, "I don't know him very well; nobody does really, but Harry is a good guy. He tries to help people in the community. Of course the powers that be don't like him very much and Harry can be a real scary guy at times. There are rumors that he was involved in some very nasty business before he came to Chicago." If Murphy pushed for more information she might get a response that went something like this, "I really can't talk about this Lt. Murphy, but there are rules we all have to live by, and there are people who enforce those rules. When I said 'The powers that be', that's who I was referring to."
It's also possible if Murphy dug out any information, it was far more fragmentary than what I described in the paragraph above, but after seeing people's hearts blasted out from the inside, being attacked by a giant magical scorpion and having Harry refuse to give her the full story of what happened, I'd be surprised if Murphy hadn't been trying to gain more knowledge about Harry and the magical world in general.