Bear in mind that Seattle is about 120 miles inland from the Pacific Coast and you have to go round either the north face or south face of the Olympic mountains to get to it.
The most direct that doesn't involve a ferry crossing is to go east from the Astoria/Ilwaco area, either on the Oregon or Washington side of the Columbia to Longview, then I-5 to Seattle. A second option is 101 north to Forks then Port Angeles. Take SR20 from 101 to Port Townsend, ferry to Coupeville, SR20 to Anacortes then Burlington and I-5 south to Seattle. You could also go north up the peninsula to Bremerton and take the ferry from there directly to downtown Seattle.
You'll have to go 15 miles or so outside Greater Seattle to find a CG. Property is too expensive in the city and close-in suburbs to use as a camp ground. Lake Pleasant, in Bothell, is probably the closest.