You ask for a "good biography of Brahms." I would recommend three, in more
or less the following order:

1) "Johannes Brahms: Life and Letters," edited and annotated by Styra
Avins (Oxford).  I know what you're thinking: this isn't a biography!
Well, it is and it isn't.  It contains extensive biographical material and
a lot of great source material -- the man's own words.  Avins challenges a
number of the conclusions drawn by Jan Swafford.  Speaking of whom . . .

2) "Johannes Brahms: A Biography," by Jan Swafford (Knopf).  Very
readable, lively, and more or less straightforward.  He speculates a bit
about Brahms' early life, but I don't think he veers too far into Maynard
Solomon, psychologist territory.

3) "Brahms: His Life and Work," by Karl Geiringer (Da Capo).
Straightforward, solid, and time-tested.

Hope that helps.

Regards,
Bob Chen
Los Angeles, CA