I went in July and the bugs were bad but not THAT bad. I started in Anchorage and went down to Valdez to see the glaciers. Quite an experience. When it comes down to hiking go to St. Elias and have them drop you off with an airplane inside the park (see here: http://www.nps.gov/wrst/) . Not for the faint of heart, though. A good place to start is Kennecott, you might be able to hook up with other back country folks here.
Another thing I really enjoyed about my visit was Denali. It is stunning, lots of wild life (foxes, bears, elk, you name it) and beautiful hikes. Just watch out for the cubs …