When I did it early May in 2013, some days were quite wet but it was warm enough that we wore shorts most days. I rode Pittsburgh to DC.
Regarding camping, camped at Rivers Edge Campground and camped in a park next to YMCA in Cumberland.
Hostels I used were Hostel on Main, and the Hosteling International Hostel in Washington DC.
Also stayed at the C&O bunkhouse which is not a hostel, it is sort of in between a hostel and camping.
I would stay at all of the above again if I did the ride again.

Our bikes and gear in Union Station in DC.

My bike after pulling it out of a box in Amtrak station in Pittsburgh.