I lived in Alaska for 15 years or so, and one suggestion is to not try to cover so many miles. Alaska is so big, you’re going to be doing a lot of driving and being too ambitious in covering the state from north to south could make the trip less satisfying than spending a bit more time in each area actually exploring and investigating what’s there.
The roads might have improved some, but when I drove the highway north of Fairbanks up to the Arctic Circle, it was a bit rough and the truckers take it pretty fast. It can get a bit boring, too. I hate to say that to a naturalist, because of course, there’s always so much to see if you really look. But if you’re just driving past it mile after mile of the same vegetation types, it is not necessarily all going to be dazzlingly stimulating. In some places, there are only a few types of trees, maybe only one species.
If I were spending four weeks, I’d do the Kenai peninsula, then the loop up from Anchorage to Fairbanks, over to Delta Junction (or Tok if you want a longer loop), and south again to Anchorage via the Richardson and Glenn highways. The Kenai peninsula alone is huge, varied, and very scenic.
You can’t drive to Nome, Kodiak Island, Adak, or Barrow, obviously as there are no roads, so if you want that experience, you would need to figure out a flight plan.
If you want to break up the trip and not always be in a car, then fly into Anchorage, take a train to Whittier, get on the Alaska state ferry and go to Valdez, and then maybe you can rent a car for the rest of the trip, up to Fairbanks via the Richardson highway, or for a shorter loop, up to the Glenn highway and back down to Anchorage via Palmer. Another option is to drive the Denali highway, which gets you into much more desolate wild country, where you can easily see caribou herds.
Do you want the Denali National Park experience? It is pretty impressive! The bus tour is the usual thing to do, but more adventurous folks can get out into the wilderness. I would make sure to take the precautions regarding bears seriously.
Be prepared for mosquitoes, too.