At this point, the Marvel Cinematic Universe is absolute juggernaut which has turned numerous actors into superstars. It seems that most actors are looking for a way in, but at the beginning Jeremy Renner actually wanted a way out. The first Avengers movie was Renner's first chance to really shine as Hawkeye, which made the fact that the character spends half the movie as a mind-controlled zombie quite frustrating for the actor. So much so that he was hoping the character would be killed off.
Jeremy Renner made his MCU debut as Clint Barton in the first Thor movie, but only in a brief role that didn't really pull much focus. If you didn't know that the S.H.I.E.L.D. agent with the bow and arrow was supposed to be a character from the comics you would have missed him entirely. We started to get to know him in The Avengers, which Renner says was also the case for him as an actor. He told a crowd at London Film & Comic-Con, as reported by the Radio Times, that he really didn't like the fact that Hawkeye spent so much of the film under the sway of Loki, as it prevented him from really getting to know his character.
It's completely understandable how it might be a bit disconcerting to find yourself playing a mindless zombie for half of a movie. As an actor, you don't even really get a chance to do what you want, you're just acting out the will of the character. In a movie which is designed to introduce each member of this amazing team, Hawkeye never really got his moment. He doesn't really get a chance to bond with anybody else in the film because, while Bruce Banner and Tony Stark are trading barbs in the Helicarrier, Hawkeye is planning an attack on it off screen.
Jeremy Renner says he would walk around the set theatrically acting like he was having a heart attack as a way to show the filmmakers he was giving them "an option" for the character. It seems to me that tempting Joss Whedon to kill your character off is just about the craziest thing you can do; it's amazing that Renner didn't get taken up on the offer.
Of course, Hawkeye did not die, and in the last couple of Avengers films he's had much more to do. We got introduced to Hawkeye's family in Avengers: Age of Ultron, making the character one of the most relatable in the MCU today. He's now glad they didn't kill him off back then.
We don't know what's next for Hawkeye, or even when we'll see it. Hawkeye has gone underground with half of the Avengers following the events of Captain America: Civil War and it's likely we won't see him again until we get to Avengers: Infinity War.