Yikes, Capital Hill could be the worst part of Seattle to run in...
If you're on the north side of C.H., you could jump on a bus to get down near the University of Washington in less than 5 minutes. The Burke-Gilman trail is a paved bike path that runs along the lake for miles, and has a dirt trail next to it in most areas. Start near the stadium and run north.
The arboretum would also be close, and offer a soft surface, but not a lot of miles to cover.
If you have time on the weekends, rent a ZipCar and head up Interstate 90 to Tiger Mountain. Miles and miles of great trails.
To the west, just north of the downtown area is Discovery Park. Lots of great trails and views of Puget Sound and the Olympic Mountains, but a bus ride there could take up to 45 minutes from C.H. Watch out for bitter urban cowboys in this area.
Seattle University is just off Broadway on C.H., and last time I was there they still had an open track, but that was a few years ago.
Enjoy Seattle!