those campgrounds on the chief joseph highway are small (not many sites) but I've never seen more than one or two campers in any of them, so if you like camping out in the wilderness, by all means, take your tent and enjoy. but do be alert to any bear activity...ask at the forest rangers office in Cody. there are some campgrounds that look nice over at the bottom of Beartooth, near Red Lodge just before you start up the side of that mountain.
one year, we stayed out of Cody at Rex Hale campground, about 30 miles out of Cody, and used the jeep and drove into yellowstone, then up and northeast out the gate at Cooke City then came down the Chief Joseph highway to Cody and back to the campground. There is elec. water and a dump station at Rex Hale.