It is not only feasible, it's easier to do now than ever before. The Alaska Highway is now fully paved (except for some construction zones). Chicago to either Fairbanks or Anchorage is about 3500 miles, and will take you a whole week of steady driving. If you decide to do it, buy a copy of the Milepost. What you may want to do is take the highway in one direction and take the ferry to/from Seattle the other direction. You will need passports.