If you plan to go this summer, you should consider making your reservations now. You need reservations for the bus and campsites. Backcountry permits are issued when your group shows up. The number of humans in each unit (the park is divided into 40-ish zones, called "units") is tightly controlled. You can't just show up and head in to the woods for a few weeks without a permit, the place is crawling with rangers (very inobtrusive, you won't even know that they are there) who will issue you a hefty fine and boot you out of the park if you don't have permit or are camping out of your zone. Permits are issued for specific units in the park, so depending on where your permit is, that's where you start your trip. Plan several trips based on the different park units since you won't be assigned a unit until you get there. The backcountry is fantastic, but it's pretty rough, there are no trails (other than game trails) so it's hard to cover a lot of ground quickly, we had a hard time covering 5 miles (with big packs) in a day (we are used to covering much, much, more) because of the terrain-- lots of bushwacking and dead ends. There are also lots of bears so make sure you have bear-proof food containers and know how to use them. If you can't get a campsite reservation, you can always camp in the "walk in" campsites by the park entrance (or at a commercial campsite outside park boundaries) and day trip into the park using the bus system.
The bus system is pretty cool, you can get on and off the bus pretty much wherever you want so you can cover a lot of terrain in a relative short period of time. The bus goes all the way to Wonder Lake (I think) which is ~80 miles from the park entrance. A bus leaves the visitors center every 15 minutes or so heading for different stops along the road, there are only a few buses a day that go all the way to Wonder Lake (it's an ~8 hour trip each way). However, the bus is NOT all that comfortable-- converted yellow schoolbusses with questionable shock absorbers. The park wildlife seem to be quite acclimated to the bus and don't run off when they hear it, so you can see a lot of critters. If you see something particularly nifty, you can get off the bus and wander for a few hours and then catch the next bus that comes along. Don't discount the bus, lots of hard core hikers take it, not just sightseers and kids.
The amount of time you decide to spend in the park depends on where you want to camp-- I think campsites are limited to 14 days and backcountry permits are limited to 30 days. There is a small general store at the park entrance and some outdoor gear stores in glitter gulch, but its not easy to resupply once you're deep in the tundra, and most mortals are unhappy carrying 30 days of food on their backs. You can leave extra food in lockers at the campgrounds. If you want to spend a lot of time in the park, maybe break it up into 2 or 3 shorter trips so you can come out and get more food and a hot shower, or try to intersperse your backcountry time with some campsite time.
Have a great time if you decide to go!