The beginning of the sentence gives us a clue: "At first I wanted..." The sentence starts in past tense, so you stay in past tense. It's irrelevant whether you are a teacher now or not. To use present perfect, you need to indicate a change in timeframes (you can simply add "now" into the sentence).

The action of becoming a teacher itself impacts on the present and future only immediately after the fact. So you could sat 'I have become a teacher' after graduating. When you are looking back on the action of becoming a teacher from the future, it becomes a past action. Being a teacher is a continuous action, not the act of becoming the teacher.

At first I wanted to be a doctor but I became an English teacher.

We can't tell what tense was meant to be used without knowing the context. It would be helpful if you included the paragraph you saw this in.