I use L.A. Totally Awesome for really stubborn awning stains (you can get that stuff at Dollar General -- it's pretty great!) and an RV awning cleaning solution that I picked up at Camping World. I have a soft vehicle washing brush that has a handle and attaches to a hose that is perfect for scrubbing the awning. Letting the soap soak in a bit to loosen the dirt and grime for about 10 minutes before using the brush is a good idea, as you won't have to scrub the awning as hard. I try to clean the awning on a warm, sunny day so that there is plenty of time for it to dry. Putting it away wet is just asking for mildew stains.
Because I have a pop-up camper, I can actually carefully flip the awning upside-down on top of the camper roof and scrub the underside that way so that both sides are extra-clean.