Ran the Haul Road from Fairbanks to Prudhoe Bay in early June (June 9-10) and the road was fine then.
Any earlier and I'd be concerned about the LOW temperatures.
You can make it a GREAT RIDE or a TREACHEROUS one. Your choice.
Some pictures from my 2012 ride can be found here:
BTW...if you make the ride...at a reasonable time...the MARION CREEK CAMPGROUND...5 miles North of Coldfoot is a good place to stay. Cheap...too. $8.
Welcome to America.
Also...it was 82 degrees F when I left Fairbanks on June 9th.
PS...When in Fairbanks or Anchorage...stay at the University of Alaska. $57 per night. Cheap. Great rooms.