After the Death of Captain America, the Marvel Comics world was devastated as the public and the superhero community grieved and attempted to come to grips with its significance and what it would mean for the world going forward. One person, Iron Man and Executive Director of SHIELD, Tony Stark saw it his responsibility to make sure the world would have a Captain America and that man was going to be... Clint Barton, the former Avenger known as Hawkeye.
Already a long-time and consistent Avenger, Clint Barton perished during the events of 'Avengers Disassembled' due to the events caused by Scarlet Witch's mental breakdown. Clint remained dead until the reality-altering events of House of M temporarily restored and killed him again, again all due to the unstable powers of Wanda Maximoff. When the world had been restored to some sense of normality, it was suggested that Clint's death was no longer certain with later issues of the New Avengers proving that the former Hawkeye was indeed alive once more. After confronting a naive and de-powered Wanda, Clint disappeared until he reappeared in Fallen Son: The Death of Captain America, a mini-series based on Elizabeth Kubler Ross' Five Stages of Grief which showed how various characters of the Marvel universe dealt with the reality of Steve Rogers death.
In the issue written by Jeph Loeb (based on an idea from J. Michael Straczynski) and illustrated by John Romita Jr., Hawkeye manages to get Tony Stark's attention but his desire to have a conversation with Iron Man about Steve Rogers death gets him knocked out by his shocked teammate. After verifying it was indeed the real Clint Barton, Tony and Clint discuss Steve's death and what they plan to do about it. Somehow Tony is able to convince Clint to take Steve's old shield out for a test run where the iconic archer has no problem wielding something that injured dozens of previous applicants beforehand. Seeing Barton has both the skills and experience, Tony offers him the mantle of being Captain America, starting immediately.
Clint barely has any time to squeeze into Steve's old uniform before him and Tony intercept Elijah Bradley and Kate Bishop, the Patriot and Hawkeye from the Young Avengers. With Tony disabled due to an EMP arrow from Kate's arsenal, Clint interrupts their escape but his debut is met by shock and outrage by Elijah and Kate. When Clint tries to question Kate's unauthorized use of Hawkeye's identity, he is surprised to learn that not only did Steve Rogers insist that Kate adopt the Hawkeye name but that she did so to honor him, not copy or replace him.
Realizing his error, Clint blocks Iron Man from arresting both the heroes and as they escape, he chastises Tony for manipulating him into being a replacement that was more to absolve him of his guilt over Steve's death rather than what was best for the American people. He hands Steve's shield back and leaves telling Tony that he'll mail the uniform back to him when he can. Tony's threats of arresting him if he sides with the anti-registration heroes still out there gets no response and Iron Man is left in the dark, in the rain, none the wiser or better.
Although his career as Captain America was a literal one-shot, it still illustrated the point that other failed Captain America's did in the past. You could never replace Steve Rogers, he was one of a kind and the best that you could do is follow his example as best as you could while forging your own path. Stark would have better success at convincing the Winter Soldier, Steve's old wartime sidekick Bucky Barnes, in becoming the next Captain America and Clint Barton would find a home with the New Avengers where he'd adopt the costumed identity of Ronin. We're not sure if Jeremy Renner's Clint Barton in the MCU would ever get such a chance but at least we know that if push came to shove, Hawkeye would have made a fine Captain America.