If you can make it work, I really enjoyed and would recommend the Chief Joseph Highway AKA Sunlight Basin Road AKA State Hwy 296 between Hwy 120 and Hwy 212 from Cody to Cooke City. This road is not in Yellowstone, it's quite remote and beautiful, and has very little traffic.
I've heard 212, the Beartooth Highway, is a must-do as well, but I haven't made it there yet.
Here's a picture from Sunlight Basin Road.
I would also recommend the western art gallery in the Buffalo Bill Historic Center in Cody. There's an OK RV park right in Cody if you need to stay there.
Look at the National Park Service web site for camping info - there are plenty of campgrounds, a couple of them may be restricted to hard-sided only (RV, no tent) due to animals. There are a lot of bears in the area, so it would be wise to start the trip with clean panniers and tent, never cook in your tent, and restrict your food storage to a limited number of panniers, which you will want to store in the bear proof containers located in the campgrounds. You are not allowed to wild-camp inside the park.
Riding inside the park can be a little scary, as the roads are narrow and the RV traffic can be heavy. I agree June is early season so it should be a little better than usual. The area right around Old Faithful is the worst. The north/east part of the park roads are less traffic-y and have a lot of wildlife.
In the Tetons, Jenny Lake has a great hiker/biker camp site right on the lake. The ride between Teton & Yellowstone is nice, and there is a lot of bike path in the Teton park.