The Aventure Cycling Association has two different routes that are relevant to this - the Pacific Coast Route and the Washington Parks Route. You can buy their maps which detail every mile of these routes.
Another coastal possibility that you could do goes from Seattle to Bainbridge Island on the ferry, then up around the Olympic Peninsula by way of Port Townsend, Port Angeles, and then around the Western edge of the Peninsula.
These coast routes are pretty well traveled by folks doing the Pacific Coast trail, if you go to www.crazyguyonabike.com
you'd find any number of journals by people who have done the coastal route.
A much hillier route would be to go more inland through the Mt. Rainier National Park. A lot of climbing but it covers some spectacular territory. IMHO it's more of an adventure than the Washington coast route. IMHO the Washington coast is not as nice/scenic as the Oregon route.
Seattle to Enumclaw (use King County's bicycle map to find routes out of the ciyt)
Enumclaw along 410 through Mt. Rainier National Park to Packwood.
Packwood to Randle.
Randle on Forest Road 25 down to Swift Reservoir (camping)
From Swift Reservoir headsoutheast to Carson, Wa.
The follow the Columbia River Gorge into Portland.
There are no services that I'm aware of from Randle to Carson; should take two days. There is camping at Swift Reservoir. If you go this route, make sure you take the road to Windy Ridge to get a view of the crater of Mt. St. Helens. It is closed to car traffic but you can get through on bike; it would be a few hours detour on your way south but well worth it.
There are two Seattle are rides, the RAMROD and the High Pass Challenge you can search for that describe parts of this route.
There is an out of print book called "Cycling the Pacific Crest Trail," available used at Amazon.com, that describes the portion of this route from Mt. Rainier down to the Columbia River.
This inland/mountain route would like take 6 or 7 days unless you push it.