When people say the PNW has year-round whitewater, they are not exaggerating. You can pretty much find any difficulty of whitewater any month of the year, as long as you include BC for late July-September.
So "what month is the best to visit and get the classics?" depends on what type of weather you want and what else you are missing by coming here.
Sure, May-July is awesome in Washington. The weather is fairly nice and the snow is melting. However, I would go to California if I had the time off then. I hear Colorado is fun around then too.
If you do want to come in Spring-Summer, Hood River is the #1 location. So many classics, tons of paddlers, and lots of other reasons to be in the area.
Leavenworth is a spectacular location in the summer. Sweet playboating on the Wenatchee. Big water V on Tumwater Canyon and fun creeking on Icicle.
SW BC in the Fall cannot be forgotten.
If I lived in Colorado and needed to quench my near-constant need for kayaking...I would come to Washington right now.
We take trips from Seattle to Hood River most often in the winter, because during the Spring there are so many other closer options.
Around Seattle right now we've got tons of rivers running. Robe Canyon, Ernie's Canyon, Carbon River, Jefferson Creek, EF Lewis, Skykomish River and plenty of other more obscure runs. It's hard to plan a trip because we do get crazy weather, like the 2-month drought that is finally over (at least here). Camping trips are not as fun when it rains all the time and days are short.