Hmmmmmm......I'm not sure buying a car is really the best option. Sure, you could find a very cheap car, but do you really want to drive a POS around in LA for a month?
Paperwork will require getting the car licensed, probably an emissions test, and insurance. The insurance part should be easy enough. We need some CA residents to chime in here. You could of course just say to hell with it, get insurance, and don't even bother getting properly registered. It will add to the adventure. You might get a chance to have a nice long talk with the California State Patrol. Just remember: Americans are instantly blinded by Aussie/British accents. They might even mistake you for James Bond. USE THIS TO YOUR ADVANTAGE.