I'm pretty sure that this argument has been around for quite some time. Each figure has a number of abilities and gadgets, yet only one can be considered that greatest of the two, and not just in combat. Intelligence, emotional background, contributions to society, and anything else all play a part in deciding who is the best: Green Arrow, or Hawkeye?
Each aspect of the two will be observed and then I will assign a point for when one of the two wins that aspect.
#1 - Combat
Before we observe the strengths of each hero, let's note one important thing... they're both EXCELLENT archers. According to the http://dc.wikia.com/wiki/Main_Page, Green Arrow, "He once shot two arrows down two different gun barrels while upside down, in mid-flip while somersaulting off a building." As for Hawkeye, not much needs to be said after his performance in The Avengers.
However, the key difference between the two is that Hawkeye is more proficient in weapons than Green Arrow. Yes, they're both trained in the art of the longbow and the sword; but Hawkeye is trained in crossbows and compound bows, and he has the potential to easily master weapons. Also, he has an almost-perfect precision in weapons that are shot or thrown. Green Arrow has indeed mastered the longbow (possibly more than Hawkeye), but Hawkeye is more experienced in weapons overall, thus granting him a point in combat.
#2 - Emotional Backgrounds
I'll admit, this isn't Batman where Bruce watches his two good, loving parents be murdered in front of him by a mindless criminal. However, both Green Arrow and Hawkeye have had traumas from their pasts that shape them into who they are now.
Different from the currently-running TV show, Green Arrow's fear to kill anything as a child resulted in the deaths of his parents due to a lion attack during a safari. Being taken in by his uncle and receiving his family's wealth, Oliver Queen, Green Arrow's secret identity, grew up as a irresponsible, drunken womanizer; until he fell of of his boat and washed up on the shore of an island, where he learned to survive using a bow that he managed to have with him. Learning to be a more responsible man, Oliver Queen made it back to the mainland and decided to use his abilities to fight crime as Green Arrow.
Hawkeye has a sad past of his own. Clint Barton, his secret identity, grew up part of his life having an abusive father with his brother, who taught him how to fight and aim. When their parents died in a car accident, Clint and his brother went through many foster homes, from which they ran away and joined the circus. During that time, Clint was trained by Swordsman and Trick Shot; and Clint later decided to fight crime as Hawkeye. It didn't start so well, however, because the police thought him to be a thief. Hawkeye then became Black Widow's partner in crime, and later chose to reform as a hero and join The Avengers.
Deciding between the two pasts is tough, because they both show that the heroes have something they are not proud of, and they didn't grow up being fully raised by their parents. However, I give a point to Hawkeye, because while Green Arrow's fear prevented him from saving the lives of his parents, at least he had a good father, because Hawkeye never did.
#3 - Contribution to Mankind
One important thing to think bout with these two heroes is that they must do more than just fight crime. Yes, they're both members of The Avengers and the Justice League, but how else do they do good?
Oliver Queen becomes the mayor of Star City and even when he is no longer in office, he still has times where he chooses to get involved with politics. Being a mayor and a superhero allows Oliver to do good on more levels than any other superhero.
Hawkeye is an agent in the organization known as S.H.I.E.L.D., where he completes missions and secret operations. Being a secret agent, he helps protect the world from the threats that we will never know about (unless you read the comics).
Between the two, I give a point to Green Arrow, because he does good not just against crime, but also against injustice that isn't sent to prison.
#4 - Costume
I believe that an important aspect of a superhero is his costume. After all, the costume is practically the symbol of the hero.
First, Hawkeye's costume. With a color scheme of blue and purple, Hawkeye's outfit brings a creative flair to the world of superhero costumes.
Next is Green Arrow. His costume is based off of Robin Hood, which is both modern and nostalgic at the same time. Also, it is quite practical.
Based on the two outfits, I give a point to Green Arrow. not only is his suit more practical than Hawkeye's, but also his is more intimidating than Hawkeye's.
#5 - First Appearance
Both Green Arrow and Hawkeye have made their appearances in the comics during the early days of their publishers. However, Green Arrow made his first appearance in 1941, while Hawkeye made his in 1964. Thus, Green Arrow recieves a point for appearing first.
The winner between the two archers is:
With the standings at 3:2, Green Arrow is the winner! Therefore, the Emerald Archer is the Better Archer.