Start with the tent manufacturer's website. They will often have specific directions there. I would first set up the tent outside and hose it off/out with plain water and let it air-dry. Then you can try washing it. If you have a "no agitator" washing machine, you can try washing it in the machine with gentle cycle, or go to the laundromat after you've got the worst of the mud off.
You will have to re-waterproof the tent, so go to your local outdoor equipment place to get either wash-in or spray-on waterproof.
Of course, the best thing is to not have this in the first place. Always set up your tent when you get home to check for damp, tears, pine pitch spots, dirt, etc. Repair or clean, and THEN roll it up and put it away for the next trip.
Source: 40+ years of Girl Scouts, plus 9 years of Boy Scouts.
By Eileen M. from Elk Grove, CA