You need a valid Passport with Valid Non-immigrant Visa and signed copies of the passport's first page, the page with the current non-immigrant visa, the page with the last entry stamp and the TM-card.
A certified letter of address from the applicant's embassy, or from the Immigration Bureau (document can not be older than 30 days).
But this is Thailand, it's possible that all this is not needed, especially as you already have a car license. Best you check with your local Department of Road Transport office.
Don't forget the doctor's certificate. I have heard of people in Pattaya getting the licence on a tourist visa but cannot confirm. The computer test is a bit of a joke, never seen anyone in Thailand doing what the correct answers indicated; that is probably why i failed the test the first time. Even before for the riding test, most people parked their bikes at a kerb with red and white lines (no parking) and no one payed any attention!
Does anyone know if I can take my motorcycle test and get a Thai Licence on a Tourist Visa? I know the proceedure you get a paper from immigration and then go to the licensing place (I have already got my Thai car drivers licence) but I did that on a one year Visa