Sir Arthur described him as "a tall, stout official ... in a peaked cap and frogged jacket." Sidney Paget's illustrations for the Strand Magazine depicts him with a full beard. Beyond this nothing is revealed about him in the canon.
Bradstreet appears three times in Granada Television's Sherlock Holmes series: The Man With the Twisted Lip, The Bruce Partington Plans (substituting for Inpsector Lestrade as Colin Jeavons was unavailable), and a cameo appearance in The Mazarin Stone. Each time he was played by Denis Lill.
He's also featured in M.J. Trow's The Adventures of Inspector Lestrade series.