The best product I've found to clean whitewalls and keep them clean is Westley's Bleche-White. It's made by Black Magic now.
Contrary to popular belief, it does not contain bleach. Wash the tire, apply the product and scrub with a scrub brush. Don't use bleach on whitewalls or, just like your socks and underwear, it will turn them yellow.
I've been using it for over 20 years.