If you can handle camping out at +10 degrees then the cheapest way to do it is to pitch your tent at Mather Campground in the Park or you could camp for free in the Kaibab National Forest which abuts the ParkIf you re gonna use the Kaibab NF makes sure that you have a DeLorme Road Atlas and Gazetteer with you so you don t get lost and at the very least a high ground clearance vehicle so you don t get hung -up on unseen rocks or logs A 4x4 isn t a bad idea either But it s not totally necessaryhttp://www.nps.gov/grca/planyourvisit/cg��Stay warm
Staying at the Grand Canyon is going to be very expensive.