Hi! We're going to be there the same weekend! We went the first week of June a couple of years ago, and the crowds were not that bad at all. It got busy in the middle of the day, but we were able to get a lot done in the mornings and the evenings.
The weather will be warm, but can be on the cool side in the mornings and evenings. Make sure you take jackets or sweatshirts. It's also usually pretty cloudy that time of year, although it rarely rains.