In Shakespeare's Othello, with regard to defining whether Othello is a hero or a villain, I believe that if we use Aristotle's definition of a tragic hero, we need not look at villainy at all.
Aristotle noted (in Greek tragedy) that a tragic hero had several qualities: he was a great (admired) man; he had to die; and, his death was the result of his tragic flaw. Othello is an imperfect man who has a flaw that he is unable to overcome. Iago uses this flaw to manipulate Othello.
There are different opinions regarding Othello's flaw. Some say it is poor judgment; others that he is overly passionate, or even that he is jealous. I believe his passionate nature is linked directly to his jealousy: and jealousy is his tragic flaw.
Alexander W. Crawford notes that Shakespeare's literary gift is...
...to place his characters under those conditions that will show the true nature of their passion and develop it to its fullness and to its fated end.
It is ironic that Othello seems to know the nature of rage, and the need to control it when he says...
He that stirs next to carve for his own rage / Holds his soul light; he dies upon his motion. (II.iii.164-165)
Even though Othello knows Desdemona (her character) and he loves her, he allows the doubts planted by Iago to worm their way into how he perceives his wife, to destroy his trust in her—which in turn picks away at his belief that she loves him and is faithful to him.
Iago plays Othello perfectly, and manipulates other characters as well. After Iago sets the "stage" and the different "players" in motion (including Cassio and Roderigo), Othello's doubt in his wife, and perhaps in himself, drives him into a jealous rage. All Othello can see is the "circumstantial evidence" found in the handkerchief. By itself, it is an innocent piece of cloth. But with the lies and innuendo that accompany this square of linen, it becomes a death knell for Desdemona.
That handkerchief which I so loved and gave thee, / Thou gav'st to Cassio. (V.ii.48-49)
Othello loses his faith in his wife, accusing her of unfounded lies that she repeatedly denies. He refuses to hear her:
"She turn'd to folly, and she was a whore", and "She was false as water" (V.ii.133, 135)
Othello kills Desdemona, blinded momentarily by his insane fury, believing the woman he loves has been untrue to him. I do not believe he is a villain. He is a great man—a Moorish general in the Venetian army. He has proved his valor. He dies at his own hand, which conveys his sense of deep remorse over the murder of his wife. He is not malicious or evil. This act does not define who he his as much as it defines what he is not: he is not a strong man when it comes to trusting his wife. I do not see proof to support that he is a villain; there is more than enough evidence, however, to prove that he is a tragic figure, a victim of his tragic flaw.