Double D -
IMO your best bet is to take I-90 and cut over to Cody, WY. Then through Yellowstone to Cooke City. You can get there on I-94 too, but you have to come South from Billings to Cody and it'll take longer. Remember, the Beartooth Pass from Red Lodge, MT to Cooke City is closed during the winter so you'll have to cut through Yellowstone at some point. The other thing is you may want to check that the roads in Yellowstone are open from Cody to Cooke City. It's been a few years since I've been there so things may have changed.
Good Luck, have a great trip!
