If you're into mountain biking, know that Outside magazine voted Whitehorse and Carcross area as the top places to mountain bike
http://www.cbc.ca/news/canada/north/outside-names-yukon-best-place-for-mountain-biking-1.1394853If you're thinking gentle pedalling along a road or paved trail, well, not so good. Many of the roads and highways don't have paved shoulders either.
Also, Yukon bears refer to cyclists as "meals on wheels". It is all too easy to surprise a bear when you're quietly and quickly cycling along a trail. Same rules apply to cyclists that apply to hikers ... make lots of noise to let the bears know you're around. They'll usually move away if they know you're in the area.
Having said that, each summer we see numerous cyclists "doing" the Alaska Highway.
Saw this cyclist (no helmet!) touring the Campbell Highway in Yukon. The black bear was chasing him up the hill until it heard us behind it. Then the bear veered off the highway. The cyclist must have had a good burst of adrenalin to help him over that hill!