The Seattle area and the Northwest in general both provide great opportunities for all kinds of backcountry hiking, camping and backpacking, from walking in a park to small day hikes to long expeditions.
Around town
The Olympic peninsula
The Olympic National Park has lots of possibilities, from day hikes and car camping to longer backcountry expeditions. There are glaciers, old forests and some rare temperate rain forest. From Seattle you can drive south around Puget Sound to Highway 101, or take a ferry across Puget Sound from Seattle to Bainbridge Island, or from Edmonds to Kingston, and drive from there. Travel time is usually two and a half to three hours from Seattle to Port Angeles, the main park headquarters. Make sure you arrive to the ferry one hour before it leaves to make sure you'll make it, lines can be long.
A good first-time place to go to in the Olympic National Park is Hurricane Ridge, which is at the end of the Hurricane Ridge Road, continuing up from the park entrance off of Highway 101. There is a visitor center at top which has restrooms and some food as well as a picnic area. The views from the top are beautiful and you can see all the major peaks of the park and their glaciers. There are a few short and easy trails around there from where you can see back to the Puget Sound and Port Angeles, and on a clear day onto Victoria, BC and the San Juan Islands. If you'd like to car-camp in the area, the Heart O The Hills campground is close by, just after crosing the park entrance. For $10 a night it offers a multitude of spots, and bathrooms with running water (no showers). Remember to take good insect repellant though since mosquitos may abound.
If you want to see a bit of everything, it's possible to drive around the whole peninsula on route 101 in one whole day. If you start on the south side you can visit the Quinault Rain Forest and see the huge moss-covered spruces, cedars and hemlocks. Continuing west on 101 you get to the ocean coast; here you can see the rugged beaches full of huge logs that have been carried out to the sea and brought back over and over again, and numerous rock formations and cool sea stacks. You can then continue the drive to the north side, where you can get beautiful views of the San Juan de Fuca straight and Canada, Crecent Lake and Hurrican Ridge.
The Olympic National Forest surrounds the national park and has additional camping locations and trails.
Other links
The Cascades
Mount Rainer
Mount Rainer National Park is about two or two and a half hours from Seattle. Lots of hiking, camping, glacier and rock climbing and cross country skiing opportunities.
Other links
The Mountaineers: "Seattle's premiere outdoor recreation organization"