Both sentences need to be corrected, but are able to be corrected in a few ways. I will provide a few options and explain the different meanings. (I will also write “boyfriend” instead of “bf” for the sake of clarity.) Original: “I break up with my bf”
 1) I broke up with my boyfriend. ← This means that it was your idea to break up and the breakup has already happened in the past.

 2) I will break up with my boyfriend. ← This means that it is your idea to breakup, but the breakup has not already happened. You will do it sometime in the future. Original: “my bf break me up”
 1) My boyfriend broke up with me. ← This means that it was his idea to break up and the breakup has already happened in the past.

 2) My boyfriend will breakup with me. ← This means that it was his idea to breakup, but the breakup has not already happened. He will do it sometime in the future. I hope this helps! Kristin https://www.italki.com/kagcafe

I broke up with my boyfriend or my boyfriend broke up with me. It depends on who made the decision. If the decision was mutual or ambiguous about who did it you can say Me and my boyfriend broke up. You cannot say "my bf break me up"

Keep going with your English. Maybe it's more importan for you at this moment.

i think the first is correct