t-bone
303 makes some really great products that are outstanding cleaners. I think the one that would best suit your spare tire cover is their fabric cleaner. Check out:
http://www.autogeek.net/303fabcontop.html
Alternatively, if you think the cover is more of a vinyl than fabric material check out:
http://www.autogeek.net/303vincontop.html
Both of these products are on sale right now.
Simply spray on liberally, let soak for a a minute of two, agitate with a short bristled scrub brush, then hose off. Repeat as necessary.
Good luck.