"I broke up with my boyfriend last month."
Don't use the に after words like 先月、今月、and 来月, because these are set amounts of time.
に can mean in, at, and to. So if you were to put に in the sentence, it would be as in. We would not use in like this in English either.
Ｘ In last month I broke up. I broke up in last month.
Ｏ Last month I broke up. I broke up last month.
You can use it after 月 in other examples.
O ５月にわかれた。I broke up in May.
The kanji for breakups would be 別れる. I am not sure if that Kanji is in the list of the first 100 that they use on Duo.
分かれる is for separating things or space.
I agree though, having Kanji makes it easier to read.