No one can predict weather - In June it can be very nice... or not. June will be less crowded and you'll see more animals with babies.... a nice time to go. Just pack warm clothes!
Explore this site thoroughly for lots of good information. This link gives the weather - 71 tomorrow, May 6 but cold nights. It also gives 'historical' numbers in the chart.
You can't boondock in the national park. You'll have to get a campground or boondock outside the park on national forest land.
Gros Ventre and Colter Bay campgrounds each have 300 sites - dry camping except for a few electric in Gros Ventre. Colter Bay RV has hookups along with Headwaters RV in the park.
Since you're planning to be in the park 5 days and if you want to drive to Yellowstone one of those days - (highly recommended), staying at a campground in the northern section of Teton NP would save some driving miles and time. Gros Ventre is a little farther south but not that many additional miles. I wouldn't recommended staying outside the park with your limited days.