As a standalone statement of fact, "I taught at this university before I became an interpreter" is the correct choice.
In a different context, the past perfect is possible:
Last year, I was offered a job at Pelham University. The salary was great and it's much closer to home. However, I really struggled to decide whether to accept the job. I'd taught at the university before I became an interpreter, and had really struggled to make any friends among the staff. They were all so different from me. Mind you, the same thing happened at my interpreting job so maybe I'm the problem! That's why I'm now a librarian. I don't have to talk to anyone!
(Job timeline: university, followed by interpreter, followed by librarian.)