You can drive to Skagway from Whitehorse, about 100 miles.
We do 4 months a year in Alaska, leaving from western WA in May. We get snowed on at least one day in May. So Late May or early June would be fine.
We cross at Sumas. The usual thing would be to get off the freeway at Hope and take 97 north up the Fraser River Canyon to get to Prince George, then left on 16 to Prince Rupert and the ferry. The alternate is take Hwy 5 over the top of the Cascades to Merritt and Kamloops, and stay on 5 north to where it intersects 16. You could take a side trip east on 16 and go thru Jasper just to do it, and then return toward Prince George and Prince Rupert. No charge if you are just going thru. On our trip we stay at Clay Banks RV in Merritt, and then at the KOA in Hinton. Nice parks, and the distance is about right.
In Alaska when you hit Tok you need a decision. If the weather in Valdez looks good, go to Valdez first. If not, go to Fairbanks.
The road (note, there really is only one road, and it loops around between Fairbanks and Anchorage, with few spurs to Valdez and the Kenai) near Anchorage will be crowded at rush hour, and Friday and Sunday evenings. If you want to see something in Anchorage, do it on the weekend, there is no one there.
Starting in mid July the dip-net season on sockeye (also called reds) get going on the Kenai River mouth and the Kasilof River until the end of the month, so lots of weekend traffic. That is also the time to try rod and reel fishing if you are so inclined. Next summer will also be a Pink salmon year, so lots of fish.
Overnighting in parking lots and gravel pits is fine. The Fred Meyer in Soldotna has a dump station and fresh water.
Seward over the 4th of July is a reservation needed, and Denali anytime. Other than those two, no reservation needed.
Many (not all) of the visitor activities are open Memorial Day to Labor Day, so staying between those dates will pretty much ensure that anything you want to do will be open. In Southeast on the Marine portion of the trip the schedule is determined by the cruise ship schedule.
If you want to try fishing, Trustworthy in Soldotna has everything you will need at decent prices. If you do the Kenai and Valdez later, there are silvers from the beach in Valdez.
Here is a site for Alaska Outdoor information Alaska Outdoors Forums
On the way home, take 16 to Jasper, then 93 south, the Icefields Parkway. I think the charge is $20 CDN fo 24 hours. It comes out on the Trans Canada in Banff, and 93 south continues about 10 miles east and will deliver you to I-90 in Missoula.
If you have not been, consider the Buffalo Bill Center of the West in Cody, WY Buffalo Bill Center of the West, Cody Wyoming
Have a great trip.