There are a couple of full hookup sites that are basically (tight) parking lots off of I 90 in Bellvue/Issaqua.... I'm not sure of their names however. They are close to the city via I 90 / I 5. I'm assuming you're thinking of UW? Take the 520 across (use the 3 person communter lane going West bound, get off on the second ramp towards Mountlake - UW exit. Take the first left after the Mountlake Bridge to get to the UW area - the first Right on 15th will get you to the back UW entrance.
There is also a KOA in south Kent that is a bit woodsey (as woodsy as you can be in an urban area)... but it is a bit of a drive in early morning traffic to the city.
You can dry camp in Marymoor park in Redmond OR the Tolt park-
Marymoor is a huge urban park with walking trails, baseball fields, velodrome wooded areas, RC plane field, dog runs, and close to the city. Tolt park is very wooded (you'll feel AWAY from any urban area) but a bit further from the city as there's no direct way to get there. Figure 40" without traffic - but it won't feel that long as it's a pretty drive. No traffic, figure 20" for Marymoor park.
These two parks would be my choices.
Let me if you need to know more stuff - I live 5 " from the UW - I'd suggest camping on my street, but the city frowns on trailers being parked (hitched) for greater than 72 hrs, and unhooked trailers get ticketed.