For hiking, I would focus on the
Wrangell-St Elias National Park & Preserve. The 11 million acres sized park is relatively easy to access by auto or plane. If all you have is a week I would recommend flying into and out of McCarthy from Anchorage. Just noticed there is now a shuttle between Anchorage and McCarthy. Using McCarthy as a base you could do day hikes or short overnight loops. You probably would not want to go in May. I was there in June. Lower valley were snow free and melting up higher. Mosquitoes were bad. When I worked in this area in 1992 and 1994 we had helicopter support. There are several outfitters that can set you up for hiking. See
http://40-mileair.com/hikingtrekking/ for the Chisana area and
http://www.wrangellmountainair.com/backpacking-trips for McCarthy area. There are several more listed on the park's website.
Given one to two weeks I would hike from McCarthy up the Nizina River over the pass dropping into the Chisana River and to Chisana where it is possible to make arrangement to fly out. Or continue onwards to Nabesna crossing Cooper Pass and the Nebesna River (might be a problem crossing or just hike up river). Fly out from Nebesna. This route established by Indians was used by miners following the Chisana Gold Rush of 1913 (the last in Alaska).