I'm not sure where the route is for the "Pedal to the Midnight Sun", but I live kinda next door in the Yukon. We also have a road that crosses the Arctic Circle. To give you an idea of temperatures, the ice usually breaks up on the Yukon River in Dawson City around the first week in May. The roads are pretty dry before that, but it's still fairly cold out in April, especially at night. By mid-May the daytime temperatures really pick up and even above the arctic circle you can expect quite warm weather. I checked the historical weather data for Old Crow, Yukon (above arctic circle) for 2010 and in late May 2010 the temperature got up to +28C. I can't remember anymore what that is in degrees F, but it must be in the 80's. Google the conversion.
I don't know much about roads or wind in Alaska, but if I was going to bike from Whitehorse to the Arctic Circle in Yukon, aiming to get there on the solctice, I'd probably go south to north and I'd leave in early June.
Good luck. Take bug spray.