I went there a few years ago on a tour with
Midnight Sun, which was great because we did quite a bit of hiking. Unfortunately they don't seem to offer that tour any more, so that's that. However, to give you an idea of what we saw, the itinerary started and finished in Whitehorse (Yukon). We first went to the old mining town of Kennicott in the Wrangell-St. Elias National Park, which was extremely beautiful. Then to Valdez for a glacier cruise, then on to the must-see of Denali National Park. A brief stop in Fairbanks was followed by a night in Dawson City (Yukon), which was a lot of fun.
The reason I went on a tour was that public transport was extremely slow and expensive, so if you don't want a tour then do sort yourself out with a car. The tour stayed in the bottom-right quarter of Alaska, but the state is so huge that unless you have several weeks it's unwise to try and see more. I went in late August, which was ideal – the midges were all dead but the days were still relatively warm.
Denali was very busy of course, but it's very easy to escape the crowds. There's only one road and it's closed to private vehicles; you have to take one of the Park buses. They'll stop at various points when there's wildlife or a nice view to be seen, and 99% of the people don't move from their seat. However, you are free to get out anywhere you want and catch a later bus onwards. We jumped out when we saw a cool mountain, hiked to the top and back down again and then caught a bus back. Only if you want to camp out overnight does it get complicated.
The next day we joined the masses and, partly thanks to my telephoto lens, got some great photos of elk, beavers, a brown bear and a load of other wild animals.
Enjoy – Alaska is gorgeous and on my "Must return to" list. There used to be a direct Lufthansa flight from Frankfurt to Whitehorse; worth looking into that one.