Halligan is an Experienced Protagonist whose dealt with Moon Logic Puzzle filled Adventure Game plots before.
It's been argued that the possible Doylist explanation for why Brent Halligan is such a callous weirdo is that he's intentionally written to be a deconstruction of point and click adventure game protagonists who go around stealing incredibly specific items from people, concoct convoluted solutions to mundane problems and have repetitive conversations about narrow and often bizarre topics of interest. The result is deeply unpleasant, slovenly conspiracy theorist Dirty Cop who is hated by all his coworkers but inexplicably bumbles into saving the world from Druids.A Watsonian explanation for how Halligan became... Halligan could be that the Skeleton Murders weren't his first experience with strange happenings and the weird logic that let him escape just became ingrained into him. My best guesses are that he was either abducted by aliens when he was young or has a career history of cracking difficult cases which built up his Ultimate Job Security until the events of the game proved to be the last straw.
Halligan is a Time Lord, has misplaced his TARDIS and is potentially regeneration sick for the entire adventure.
Instead of being a renegade though he's on a mission from the High Council, potentially even the CIA, to prevent the Inheritors from tying knots in the web of time with their druid magic. His background cameo in The Moment of Silence is a later adventure after he'd started traveling with Melanie as a companion.