The Goat Rocks is an excellent backpacking trip. You can hike it a number of ways, I think you would enjoy hiking from the Mt. Adams side along the Pacific Crest Trail and traveling the 41 mile one-way trip to White Pass (car shuttle required). It is quite spectacular if the weather is good. If the weather is bad, you will be denied some fantastic views and also be hiking through rather exposed terrain for past of the trip. Making this all the more challenging is the fact that the trail is narrow on a knife-like ridge during some of its most majestic sections.
You will have enjoy excellent views of Mt. Adams and Mt. Rainier, as well as many beautiful miles of walking. If you get done early, it's not a huge issue - the southeast entrance to Mt. Rainier National Park isn't far away. You could drive through the park on your way back to Seattle.
May I also suggest the Spider Gap-Lyman Lakes-Cloudy Pass--Image Lake - Buck Creek Pass route described here. It takes you through the Glacier Peak wilderness just to the northeast of Seattle. Terrific. You can even double back once you get to Image Lake and rather than hike back via Buck Creek pass, proceed down the Pacific Crest Trail toward Steheiken, only accessible via boat or foot. This would be more involved logistically, but at the end, you have a rather religious experience at the Stehekin bakery on Lake Chelan and could could travel via Ferry to the town of Chelan. I can provide more information on this, it has some logistics, but is also very rewarding.
I would really suggest a backup plan in the event the weather turns. If the weather isgood, the west side of the Cascades would cover any number of trips, including those in the Alpine Lakes, the Glacier Peak and the Goat Rocks. But if the weather is bad, escaping to the east side of the range really can help make your trip much drier.
On that note, the most accessible from Seattle is the Teanaway Region. There are a number of great loops out here with lake-side camping. We hike out here often, especially in summer and fall. September is one of my favorite times with the change in colors.
If you want more information, just PM me. I will be glad to help you out. But as Dale pointed out nwhikers.net is a fantastic resource as well - the people there have hiked just about every trail imaginable and can make really informed recommendations.