Prince George is the largest northern BC city, with good connections east to Jasper via 16, and south via 97 to the TransCanada.
The nearest town in Yukon is Watson Lake, 800 miles, 15 hrs drive according to Google Maps (via Dawson Crk and the Alaska Hwy). The highway stradles the BC/YT border for a ways here.
The alternative is the Cassier Hwy, west and then north, 760 miles, but a bit slower.
The nearest town in Alaska (from Watson Lake) is Skagway, 320 miles, 7hrs.
The other town with road access is Steward/Hyder. That's 430 miles, 9 hrs from Prince George. (Watson Lake to Stewart is 400 miles, 10 hr).
So the shortest way to touch both Yukon and Alaska from Prince George is the loop to Steward, north to Watson Lake and back, 1700 miles, 35 hr of driving.
Years ago I touched on much of that in a 2 month trip.
Chicago to Edmonton; Alaska Hwy to Whitehorse; north to Dawson and on to Inuvik in the NWT; Back to Dawson and west to Alaska. By then we were 21 days into the trip.
The return was via the Alaska state ferry down the Panhandle, and on to Stewart (that route is no longer used), then east to Prince George and Jasper. That was about 3 days from ferry to the National Park.
Unless you intend to go all the way to Alaska, or at least really see the Yukon, I don't think it is worth the drive to try to 'touch' on the Yukon and/or Alaska. It's just too far of a drive from anywhere centrally located.
If you have time and the money (for the ferries) I think a loop via Prince Rupert would be more interesting (with a possible side trip to Stewart/Hyder.)
Vancouver to Vancouver Island, north to Port Hardy, ferry to Prince Rupert (a day trip in the summer), Prince Rupert to Prince George, south to Cache Creek. Then several choices - several routes south back to Vancouver, or east and south to see the rest of southern BC.