We have been visiting WDW during early June for the past 4 years and we have found the crowds to be very manageable. It is busy but not as busy as it gets later in the month. We are usually able to see and do everything we want with a little planning and FastPass Just be aware that the first Saturday in June is Gay Day at MK and it can get pretty busy there. (We don't visit MK on any Saturday in June so I can't really speak from personal experience.) The crowds can't be too bad because we will back June 4-15
We were there the last two weeks of June and the heat was awfull. It was crowded but the heat made it worse. Seemed like everyone was looking for a shady spot.