Motorcycle license in Thailand
An international license will be OK. Usually nobody will check you for license and the renters do not either. A couple of things to be aware of:
Check the bike completely prior to taking it away from the renters for any scratches and damage, if you have a digital camera, take some pics of it and the damage so you wont get tagged for it when you return.
It's always easy and fun to let the road and driving get away from you when you are out. "Be careful" If you get into an accident you are just asking for a lot of trouble with the renters and the police and it will probably be costly. Many parts of Phuket can be really busy with traffic. The open raod with less traffic can actually be more dangerous, people get going too fast with an unfamiliar road and crash. Being a Farang, police sometimes insist you wear your helmet and will fine you when you don't. You have not posted where you are from so I don't know what side of the road you drive on! Be aware of traffic regulations, even when nobody else is!
Aside from all of that, you can see a lot more from a bike and have a lot of fun.