They are getting a lot of snow in the Cascades this winter. It was a very snowy year the first time I did the Northern Tier west to east. We crossed Rainy and Washington Passes at the beginning of June. The rain we encountered on the way up turned into snow as we got higher (nothing stuck to the ground), and there was snow plowed six feet high along the side of the road. This was taken near Washington Pass:http://www.flickr.com/photos/davez2007/5246696015/
Crossed again the following year at the same time. Less snow, but there were some light snow showers. As noted, you will have to decide whether the possibility of encountering such weather is for you or not and adjust your dates accordingly. Personally, I am glad I went when I did. Made for an epic ride, and the tourist traffic was nonexistent. Note that Winthrop, Okanagan and Tonasket will likely be warm and dry in early June. We had snow again in Republic and on Sherman Pass. Colville, Ione and Sandpoint were also chilly and damp. I agree that I would not like to be in some of those places in August due to the heat.
As noted, prepare for a long stretch without anything. If I remember correctly, the climb from Mazama to Washington Pass is about 15 miles. Then you go down for about 3.5 miles and then up 1.5 miles to Rainy Pass. From there, it's another 30 miles (if not more) to Newhalem, where there at least was a store. If it’s no longer there, you will have continue on to Marblemount. Most of that 35 miles is descending. It’s a beautiful day.
While not as long or isolated, the climb up the east side to Loup Loup Pass has sustained sections of at least 8%. Sherman Pass is a good climb, but not killer. The east side of Wauconda Pass heading west from Republic is not bad. Earlier on, there is a very steep, twisty section of road heading west from Ione. It looks like a vertical wall on the map’s profile. Fortunately, it’s probably only a mile or so. Unless things have changed for the better, resist any temptation to take U.S. 2 out of Sandpoint. Little to no shoulder in places because of rock cuts and traffic, including logging trucks. Follow the route proper.
I found the route to have good services, including adequate camping in most places. If the same family still owns the Winthrop KOA, they are very nice people and give cyclists a deal. There is also a new cyclist-only camping area that opened last year between Winthrop and Mazama. It’s called the barn or something like that. Howard Miller Steelhead Park in Rockport has Adirondack shelters.