Don't do it. As others have said, roads may be icy and possibly snow covered in areas, and virtually nothing will be open. April and May is when the snow melts, flooding tends to occur, and everything is muddy.
Plus, the new RV you'd be picking up in Chicago will NOT have winter tires on it. Despite what they say, "all season" tires are really three-season tires, and are not good on snow or ice. Coming from a desert climate, you presumably don't have experience driving on slippery surfaces, and RVs are very lumbering vehicles.
The best times to visit Alaska are June and July, and then March if you want to do some winter activities. If you really want to see Alaska during your April window, consider instead taking the Alaska State Ferry up from Bellingham, Washington (no need to bring a car), or fly up.