I was thinking of Henry V before I read all of these posts, and 9 beat me to it but I am still going to use it because he is very inspirational. All of the other heroes are tragic heroes, and seriously flawed. Henry V, on the other hand, was a war hero.
For pure hero, I would have to pick Henry V. His stunning victory at the Battle of Agincourt saved the English army and crushed the French power. Fighting in the Hundred Years War was halted for a time until the heroic king died and he was succeeded by his baby son.
As for tragic hero, I would have to choose Hamlet who succeeds in his quest at great expense.
By definition, a hero is always good. That being said, we have to remember that tragic heroes are tragic because they have flaws--or at least one flaw--equal to or surpassing their strengths. To that extent, I pick Hamlet because his flaw is primarily being too moral to take decisive vengeance. It's true he doubts himself and does not do what he purposes to do. What's also true, though, is that he understands the eternal consequences of death and murder. I respect him for that.
I suppose this depends upon how "greatest" is defined. If we're talking about personal character, I would have to go with Hamlet. He is a young man who is thrust into a situation beyond belief and in which he has played no part. While his uncle is busy killing his father and marrying his mother, Hamlet is away at school studying theology. He is a serious person of conscience, and one of strong religious beliefs. When he returns to Denmark, Hamlet is tormented because every part of his identity is challenged: Son, Danish prince, Christian. He wants to do the "right thing," but there is no right thing for him to do, considering who he is.
That said, Lear is considered by many to be Shakespeare's greatest hero because of the very depth of his tragedy. Although he is a vain and foolish old king who brings his fate upon himself, Lear's human suffering surpasses that of any of Shakespeare's other heroes.
My answer is based purely on personal taste and opinion since the word "greatest" has many implications. Othello could certainly work as Shakespeare's greatest hero. He, in many ways, is a victim of setting. As a Moore living during Elizabethan times, he has no one to trust. He has worked tirelessly to become a respected military leader and has legitimately won the love of a compassionate woman.
His downfall and choice to kill Desdemona are his responsibility, yet as a reader I feel the more sympathy for Othello than for any of Shakespeare's other tragic heroes. I believe that this is because while he allows himself to be manipulated by Iago, so does everyone else--very few of whom face the end results that Othello does. Additionally, all his life is used by others until he finds Desdemona and Cassio who clearly care for his well-being, but he is not secure in their love or friendship because of the way he has been treated by all others. That insecurity allows Iago to carry out his plot against him, and Othello never gets to witness justice occurring against Iago.
A tragic hero is always great. In the Shakespearean tragedy the central character - the hero - is always conceived as an outstanding personality, despite its faults and negative sides, which inevitably make him more human and capable of arising sympathetic feelings; and this can happen even when they reveal their worst and darkest nature. For this reason Othello and Macbeth could be looked on the "greatest" heroes. They are brave, successful, respected and theirs are promising lives and careers. Nevertheless they succumb to mean passions: jealousy and ambition. The blind Othello fails in understanding Desdemona's love whereas the desire for power brings Macbeth to upset his spiritual balance. Both are great - as tragic heroes - in their characterization as victims, as human beings who are driven by Machiavellian forces.
In this sense the readers, as well the audience, share their suffering made greater and deeper by the understanding of their passions and the reasons which made them victims.
In my view two close contenders for the 'greatest' would be Macbeth and Hamlet. I haven't read about any other killer comparable to Macbeth. Macbeth is a wonderful case-study in self-dividedness: both 'fair' and 'foul', ambitious and conscientious, a bloody killer and a poet with rare flights of imagination, a man who inflicts tortures and tyranny & also suffers from extreme self-tyrannising states. As I intend to choose him to be the greatest, there comes the Prince of Denmark, Hamlet: the young, scholarly protagonist faced with the problematic of evil, keen to avenge the murder of his father but enmeshed in a profound dilemma--'to be or not to be, that's the question'. Ya, that is also the question for me.
Thanks for your helpful insight!