I originally classed Billy as a Ravenclaw until I started writing why he wasn't a Gryffindor and changed my mind. Billy craves knowledge and is an incredible perfectionist, particularly in academics, but I think other qualities take precedence.
While Billy considers himself clever, I don't really think of him as particularly resourceful or cunning -- he's far more likely to confront someone directly than try to undermine or undercut someone. Slytherin
Hufflepuff While Billy's not unwilling to work hard, he'll take the easy way out if he can -- and he can be intensely impatient when things don't fix themselves. I'd count him as exceptionally loyal, but ...
Billy is the Hermione Granger sort of Gryffindor, an uptight perfectionist who seems like they ought to be in Ravenclaw. However, this one is about when push comes to shove -- Billy's underlying strength is kind of a dark horse, something that doesn't come up often. He'd die to save someone he loves. He'll step up to the plate and volunteer to train as a Sentry, because despite being afraid of his powers he knows he has something to offer. He'll make all manner of rash decisions to protect people he cares about, he'll lash out in aggression but he desperately wants to do the right thing. He's vain and he's prissy, but under the right circumstances Billy is selfless and recklessly brave.
And ever since having his world opened up by going on that mission with Dean, Billy craves adventure. He wants danger, he wants excitement, he wants to live a life bigger and more exciting than himself.
The Scarf of Sexual Preference says: Please, Sir, May I Have Another?