If you want to become a driving instructor, you will have to meet the minimum requirements set out by the DSA. Since the one that's of most concern to you is about convictions, I'll skip the other criteria. When you apply to do the driving instructor course, you shouldn't have been disqualified from driving within the last 4 years. This means that any driving ban which was in effect during that period would prevent you from being accepted because you would have been disqualified.
If your drink driving ban ended 4 years or more from today, then you are fine to apply. However this does not guarantee anything. Each application is scrutinised to check the ability and character of the person looking to join the driving instructors register. So someone with a DR10 could have been accepted but if you could be rejected if you have any criminal records. The best thing you could do is call a driving instructor training centre and see if you are fit to join the programme by giving them all the details they need about you.