We have been to Yellowstone twice now, the first time we camped the entire time at Fishing Bridge. So we had to drive everyday, alot(hours).
The second trip we learned our lesson, and spent 2 to 3 days at Fishing Bridge, then the camp ground N. Yellowstone(don't remember the name but it's still in the park), and then the campground that's near Old Faithful. Keep in mind that Yellowstone is huge, and so I would definitely recommend doing what we did during our 2nd trip. Also, I think fishing bridge is the only campground with hookups, but if you are in a Class C that shouldn't be a problem. I know you will enjoy your trip, Yellowstone is one of our favorites.