Two characters with near perfect aim – one villainous and one heroic – going at it against each other. Cool.
Bullseye was an assassin, hired by the Kingpin to kill Nikolas Natchios, who was notorious for having great aim and his ability to use just about anything as a lethal projectile (such as a playing card, which he used to slit Elektra’s throat, for example).
He was also a capable hand-to-hand fighter, but it was certainly his aim that made him most dangerous.
He killed the aforementioned Nikolas Natchios by throwing Daredevil’s own billy club in to his chest and killed numerous other henchman with a series of different and very unlikely projectiles.
During the credits at the end of the movie, Bullseye, having been moved to a prison hospital and severely bandaged up after his confrontation with Daredevil, was being irritated by the constant buzzing of a fly around his face.
He struggled to reach a nearby syringe, but managed to do so, and was shown to still have his perfect aim despite his injuries when he uses it to impale the fly on the wall.
However, he was shown to be capable of missing when Daredevil dodged one of his throws.
Hawkeye, on the other hand, displayed some outrageous skill and had some pretty high-tech weaponry on his person.
He used a specialised bow made specifically for his arrows. It came with a quiver to house the numerous arrows, was mechanized and could swap arrowheads such as standard projectile arrows, explosive arrows, arrows with electrical devices attached to them or grappling hooks.
Simply pressing a button on the handle grip of the bow made an arrow combine with the selected specialised arrowhead. It was also foldable, as seen when Hawkeye was in Loki’s underground base when he’s retrieving it from his case.
Hawkeye is extremely proficient in its usage and can be accurate with his aim even when he’s not looking at his target, such as when he took out a Chitauri flier just by hearing it behind him.
The fact that he can use it as well as most people use automatic weaponry is testament to his skill and he was clearly a proficient hand-to-hand fighter as well – taking out several Chitauri soldiers with relative ease.
He should do quite well here, all things considered.
Winner: Due to having better equipment on his person as standard and by virtue of never having been shown to have missed – Hawkeye.