We went last year June 10 through the 23 rd. it was crowded but as others have said, if you stick to early morning, take a break in the afternoon, and come back in the eve it is not too bad. Make sure to book your FP as soon as your window opens and coordinate with rope drop, if you can.
As far as the heat, I was surprised. It was not bad. We even had two nights I could of used a light jacket. We usually go in Sept and I am used to that wall of heat that hits you the minute you leave the plane airport. June was sooooo much nicer temp wise compared to Sept. That is why we are going this year June 6 - 19.
Actually now that I think of it I agree with pumpkin and Spiderann. No one should go the first week of June. I heard it will be horribly crowded this year and the hottest it's ever been.