We usually go to Disney this time of year. Park attendance on weekdays is somewhat busy, but manageable, especially compared to later in the summer. The exception being the first Saturday of June at the MK, there is a GayDay event, and the park is crowded, however, the MK is crowded almost every Saturday so I don't know if crowds are up specifically because of the event...there are sizeable conventions and gatherings at Disney year round.
You may find the weather HOT, depending on what part of the US you are from...living in Texas, it is just the norm for us to have hot June weather and Florida is no different.