Look into a fall trip, which in Alaska is the end of August, early September. Get into Anchorage, then head up to Denali. It's about 4-5 hours in a motor-coach to get to Denali and a little slower on the train. This transportation is available through most cruise lines as an "in-land" excursion.
When you get to Denali, head out in one of the buses through the park. It's the only way you can get deep into the park. Arrange a night's stay in Kantishna. A bus can take you through the park at a leisurely wildlife viewing speed to Kantishna (about 90 miles in). For best wild-life viewing, hop a morning bus. The lodge there has good food. Spend some time hiking and enjoying Wonder Lake. You also get good views of the mountain (Denali) from there. If you want to bring or rent a bicycle, you can use the camper buses to go into the park. They will drop you off wherever you want but you must keep bicycles on the road (dirt road). You can also flag down any camper bus and get a ride back the other way if space is available, should you choose to end your ride early.
There are other side trip adventures as well. Rafting, obviously hiking, bike tours and helicopter tours come to mind. My company operates the park in conjunction with the native peoples, most notably the Doyon tribe. Even that late in the year, it is daylight until 10 or 10:30pm and light again very early (4:30 or so if I recall correctly).
Everything that Alaska is promised to be - it is. This is wide open, wild country. It is a place for the adventurous of spirit. I spent two weeks last fall working up there and will be back up at the park for a month in May. I'm not just looking forward to it, I actually dream about going back...
angela