I suggest that staying in the Forest Service campgrounds that surround Yellowstone is vastly superior to staying in the park.  The NFS sites are relatively primitive, but smaller, quieter, and generally the choice of the locals.  The in-park campgrounds are enormous, crowded and noisy.

I was there with my wife and kids this summer.  We stayed a couple of days in the Newton Creek NFS campground about 10 miles east of the east entrance.  I'm not sure if this was one of the victim areas of wildfires this year or not.  The campground rests next to the Shoshone River, with 300' cliffs and rock formations on the other side.  Beaver, deer, black bear and mountain goats called the other side of the river home as well.

As for getting there, we took Hwy 16 from the Black Hills in our '85 Wasserboxer.  Be ready to spend a good bit of time in lower gears if you go that way.  But with all the beauty, there's no need to be in a big hurry getting across the Big Horn Mountains.