An essay on Brahms has to include something about his relationship with the Schumanns: Robert and Clara. I think there is an entire soap opera there. Also you could touch upon his late blooming in composing his First Symphony and his reworking of his First Piano Concerto from a symphony to a piano concerto. Just some thoughts to help you out.
I am fascinated by that. As mentioned before, I think there may have been more behind closed doors!