Can I still qualify for a teaching position abroad with a criminal record?

During the hiring process, you undergo a background check and that will pull up your criminal record. Hiring decisions are made at the discretion of the school, but a criminal record is not immediately disqualifying in every case.

It's best to do further research into the country you're interested in teaching in to find out if they will hire teachers with a criminal record, or a teacher who has been convicted of a specific crime (as their decision may vary according to the type of crime committed).