C.W.
I would use a sponge and some woolite. Take it outside and spray it down with the hose and rinse it off that way. If it's really natsy I would carefully remove it from the frame. I would take notes to make it better to put back on. Use a larger but sharp needle and some button weight or upholstry thread and carefully hand sew it back on. Wash in the washer but dry it back on the frame incase it tightens up. Which it might do in the dryer. I am a seamstress if you cannot tell.