May 6's Thor
Not a lot. Renner's appearance in at least one scene (with lines) in Thor was only recently confirmed. Hawkeye also showed up briefly in a preview clip just long enough to aim the character's trademark bow and arrow — Paramount later confirmed that the unrecognizable character was, in fact, Hawkeye.
Renner as Hawkeye has been in the works for a long time, with the actor rumored for the role since 2009. Screenwriter Penn admitted to Collider in mid-April that Renner playing Hawkeye was his idea: I think he is in The Avengers because I pushed them. I said, 'We've got to get Jeremy Renner for that part!'"
Last August, Whedon revealed that Renner was already "in [Marvel's] sights" by the time he came on board the project.
Like Black Widow, Hawkeye works for S.H.I.E.L.D. and, in the comics, the two characters become romantically linked. Renner hasn't revealed whether that will happen in the movie, though he has said the two are "kind of a team".
He's described the character as "very physical" and, obviously, is training in archery. Renner has also revealed that his character will not appear in the purple costume worn in the comic books.