It's actually not a requirement to bring a criminal background check document (CBC) to Thailand, although some employers may ask to see one.
The vast majority of employers have realised that generally teachers arrive in Thailand without the CBC and getting one from their own country or state is just too time-consuming and too much hassle.
The real problems can occur though when a teacher goes to a Thai embassy or consulate in a neighboring Asian country to apply for a non-immigrant B visa on the basis of obtaining work as a teacher. At some embassies and consulates, they will ask to see a CBC. Not all of them but some.
Getting yourself a CBC while you are in your own country is always a wise idea. There's every chance you won't need to show it but always be prepared for a first time.
The fact that you've been in prison for the crime committed (information withheld) won't matter.