Hey JD,
Check out my trip report from last May; a bit earlier than you plan to be out, but you could expect similar conditions in June; but in a light snow year, you could also have much less snow. All depends on the snowpack.
http://www.14ers.com/php14ers/tripreport.php?trip=11767&parmuser=nyker&cpgm=tripuserSnow on the Main trail might be gone by then or it may not be.
As far as gear, you'll need to iron out more details in addition to a coat and sleeping bag and fine tune your system ahead of time to help increase the odds of a more comfortable trip; With that said, figure on having 3-4 layers plus a shell on top, and then a pair of hiking pants, possibly thermals and shell paints for the bottom. A lightweight down sweater/jacket with a hood would be helpful around camp.
You don't mention where you are coming from or experience, but I'd suggest that you read all the posts and helpful info on this site and post any other specific questions you might have.
Good luck and be safe.