Air compressors are a fantastic way of clearing out a chainsaw air filter and probably the most popular way. However when an air compressor is on too high then you do run the risk of damaging the filter. If you are using an air compressor then blow at around 40 psi from the side to avoid damage.
Vacuuming does clean out the air filter and if done before using the next method is an extremely effective way of cleaning an air filter without using an air compressor.
Warm Soapy Water
Dish soap and warm water at the end of the day, then let it sit overnight to dry. You would be surprised how much stuff comes out, even after the air compressor or vacuum.