It's Okay, I'm an Avenger
One of the arguments that I always find most interesting regarding Clint Barton is that he shouldn't be a hero given that he's an unenhanced hero without any special armor or weapons to protect him. I find that the argument shows just how little people know about Clint. Clint Barton is more than just able to keep up with the other Avengers he has defeated many of them and saved them more than once.
In Avengers Annual Vol. 1 Issue 2 Hawkeye and Black Panther (though mostly Hawkeye) defeat the Hulk. Hawkeye starts off fighting Iron Man while allowing Black Panther to take on the Hulk. As with the first time he fought him he is able to easily defeat Iron Man using one of his specialty arrows. In this case it is his acid arrows. Once Hawkeye takes down Iron Man he goes to help Black Panther and is able to take down the Hulk by knocking him out with a sonic arrow. Hulk then turns back into Bruce Banner following the battle.
Clint has also beaten Bruce Banner's cousin, Jennifer Walters or She-Hulk. In Avengers: West Coast Annual Vol. 1 Issue 2 The Avengers are again forced to fight one another. She-Hulk initially appears to have the upper hand when she manages to break Clint's bow and take his arrows early in the fight. With no weapons left Clint uses his fast talking to distract She-Hulk while he lures her into a trap. From there he makes a new bow from what he has on hand and defeats her with a barrage of explosive arrows and knockout gas arrows in order to claim victory.
As mentioned above Hawkeye has beaten Iron Man on more than one occasion. The most memorable time was during their initial battle (and Clint's first appearance) in Tales of Suspense Vol. 1 Issue 57. In this comic Black Widow tricks Hawkeye into believing that Iron Man is his enemy. Clint initially confronts him with an arrow laced with an acid that rusts Iron Man's armor. In their second battle he is ultimately forced to retreat but not before catching Iron Man in a net and even shooting an explosive arrow his way. If not for Black Widows injuries he would have no trouble taking Iron Man down again.
Hawkeye shines most when he is taking down enemies, rather than the rare occasions when he is forced to fight his allies. In Avengers Annual Vol. 1 Issue 16 Hawkeye and the other Avengers are forced to fight the Grandmaster in a dual to the death. The Avengers are divided into several teams where they are forced to battle some of their most famous foes. While most of the Avengers are either defeated or succumb to injuries caused by unlikely mishaps (such as being impaled on a stalagmite) Hawkeye and Captain America manage to survive. Rather than continuing to fight their way out Clint manages to beat Grandmaster in a game of chance and save their allies. Of course for Hawkeye it was a calculated risk, not a game of chance since he was only counting on Grandmaster not noticing him cheat.
Another comic that shows Hawkeye's talents is Avengers Vol. 1 Issue 174 where Hawkeye is the only Avenger who is able to avoid being captured by The Collector. Rather than escaping solo he risks himself to save the others. Clint faces The Collector at his strongest when the aged Immortal is using not his gadgets but his own cosmic power against the archer. Hawkeye continues to march forwards however and he strikes The Collector with an electro-arrow, forcing the old immortal to release his friends and admit defeat.
Despite proving himself time and again some people still write off Clint Barton's victories as flukes or one time things. I think to some extent they could be right, after all any hero can triumph under the correct circumstances. If that was all these amazing feats were though I doubt he would be as highly respected and powerful as he is. After all, Clint Barton is one of two completely unenhanced humans (at least with regards to strength) who can wield Captain America's shield the way Cap himself would. This is why when Steve Rogers died during Civil War, Clint Barton was the first person Tony offered the mantle of Captain America to. Of course, Clint turned him down (in part because of Kate's intervention) but the fact that he was the first person considered for the role speaks volumes to his abilities and the respect that other Avengers have for him.