Once the suede is dry and clean of dust, apply a liquid silicone to the surface, and work it into the suede with a damp sponge or toothbrush. You can find special suede brushes, erasers and degreasers, but they are not usually necessary.
Using special leather cleaning products, you can actually wash suede leather clothing in a washing machine. Just make sure to turn it inside out, and wash on a permanent press cycle, which uses cold water to wash and cold water to rinse. Air drying is the best, though suede can soften if dried on the air dry (no heat) setting on the dryer. Machine washing is not recommended on stiff suede clothing, such as shoes or hats.