I live in Washington and here's what you'd probably want to see and do in the Pacific NW:
You can see a few of my WA pictures here:
http://picasaweb.google.com/jgrothe
Mt Rainier NP - the most prominent mountain (looks he biggest) in the 48 states. Tons of huge waterfalls, wildflower meadows, glaciers, and breathtaking views. Keep in mind these mountains are big - they look bigger (when measured from base elevation to peak) than anything else in the 48 states.
Olympic NP - this park is extremely diverse. There are a couple rainforests as people noted (look for pictures online). There are also high mountains similar to the cascades (there's a visitor center you can drive to at hurricane ridge).
My favorite part of Olympic NP is the coast - it is, to my knowledge, the most undeveloped coastline in the 48 states. The PNW coastline (WA and northern OR) is extremely rugged and beautiful. Here I would definitely recommend backpacking (it's only about a 3 miles hike to the northern beaches and you can rent gear at REI). If you go on a weekday there will probably be few enough people that you don't even notice them.
I've been backpacking and hiking (50+ different trails in 4 years) in the PNW. Every trip was extremely beautiful, but there was just something different about camping on the Olympic coast. It's awesome pitching your tent right near the water, hoping that you're above high tide line (okay it's pretty obvious where high tide line is). It's amazing watching the sun set over the ocean, then building a huge fire with driftwood while you listen to the crash against the shore. One time I was in a section that had very high rocky cliffs right above the beach. I picked my campsite such that when high tide came in, I was completely blocked in against the cliffs. I had my own 100 foot wide private beach surrounded by cliffs.
There is also North Cascades NP, one of the least visited national parks in the country. It is very beautiful and is a great place to backpack, but Rainier and Olympic NP would be higher priority for sure.
I think this would be doable in 4-5 days. At both parks you can see a lot without going more than a mile or two from your car, except that I would try to backpack the coast.