I do a fair amount of June hiking in the Catskills, and I have never encountered biting black flies. Hopefully others who have will post their experiences. From reports that I have seen the season in the Catskills is generally late May through June, perhaps a 1-2 weeks ahead of the Adirondack High Peaks. It varies widely based on weather. As everywhere else they hatch out earlier at lower elevations and later at higher elevations. They can be bad at any level - just on a different schedule. There are wide temperature variations in June -- so it is difficult to predict how bad the black flies will be for you backpack. They tend to be out on warmer days, while cooler days may have no bugs of any kind.
Even if you do not encounter black flies the gnats, mosquitos, deer ticks, and pesky (but non-biting) May flies can be bothersome. So plan to have repellent with you. But in general the bugs in the Catskills aren't as bad as in the Adirondacks and other northern mountains.