For the last two years I've gone the second week of June and probably won't again if I can avoid it. It's hot, muggy, a bit crowded and very rainy. This past June it was really more than just rainy, there were heavy thunderstorms with lots of lightening and downpours. We walked in rain up to our ankles getting to our PS at Teppanyaki (Japan World Showcase). Never got to see Fantasmic due to rain.
We are WDW pros and know how to use FastPasses, get there early, leave in the afternoon for the pool then return at night. But even getting to the parks at opening we would be soaked in sweat just waiting for the gates to open. It was a bit crowded but doable. I can't say we had trouble with lines or anything, it was mainly the heat & humidity that just drained us. And it's just so hard to plan around all that rain. Rides would shut down, hard to get to PS's, huddling under shelter with other wet ponchoed folks.
Like I said, we'd do it again if it was the only choice, but there's just better times to go.
If you can hold off until the end of October that's a good time to go. It's still warm, crowds are lower, we had no rain but it's possible it could still rain I guess. Plus, the Food & Wine Festival is going on and that's always fun for adults to do.
We really enjoyed our trip at the end of January, first of February last year too. It was a little cool for us to swim but we weren't interested in swimming anyway. Others were swimming though. The crowds are nonexistant and perfect touring weather as far as temps go. I hope we have the same luck this year.