You would do quite well with two nights in Anchorage, one day of travel between Anchorage and Fairbanks, and then either two nights in Fairbanks or one night in Fairbanks and one at Chena Hot Springs. Both cities have enough attractions to soak up a couple of days in the winter months. You could also add an extra night at the Alyeska Resort in Girdwood, about 45 minutes south of Anchorage.

Concerning the drive, I would strongly not recommend doing it in a rental car in the winter. If you don't want to fly up, the Alaska Railroad is a fantastic trip, although it only runs on the weekends in the winter. Goes north on Saturday and south on Sunday.