2 of the best sharpshooters in the Marvel universe. Bullseye, a man who can kill you with a paperclip. Bullseye is lethal with any weapon. He has an unmatched pinpoint accuracy that can make him a threat with any weapon. This makes him the perfect assassin. He never misses.
Hawkeye uses mainly bows and arrows, but giving the option he is probably as lethal as Bullseye with anything. What makes me think that Hawkeye would win this is because Hawkeye is the better fighter. He is trained in a multitude of fighting styles and his arrows are custom fitted with explosives and other attachments. This makes Hawkeye the better option.
It would be an amazing fight. Both men would have a standoff, just waiting for the perfect shot. Both men are so disciplined in their approach that a winner wouldn’t be decided for a while. When the fight gets close is when it gets interesting. Like I said, Hawkeye is the better fighter, skilled in many forms of combat. It would be tough for Bullseye to take him on in a one on one fight.