We don't know for sure exactly why Iago hates Othello so much and wishes to destroy him. All we have are Iago's own explanations, and given that Iago is notoriously devious and untrustworthy, it's best not to take anything he says at face value.
Nonetheless, there does seem to be a deep-seated insecurity underlying the reasons Iago gives for hating Othello. For instance, he says he hates the Moor because he was passed over for promotion in favor of the less-qualified and experienced Cassio. There's more than an element of class insecurity here. Iago's not as high up on the social ladder, and he deeply resents the fact that the upper-class Cassio has taken what he believes is rightfully his.
Iago also claims that his wife, Emilia, has been cheating on him with Othello. There's almost certainly no truth to Iago's suspicions, but he's so insecure in his marriage, as in much else in his life, so unsure of himself, that he's ready to believe that he's being cuckolded. Once again, we're dealing here with Iago and the alternative universe he's constructed for himself. His hatred of Othello has little or nothing to do with reality; it's all about his paranoia, insecurity, and resentment.
The question of why Iago hates Othello so much is one that has occupied scholars for centuries, for although Iago certainly does offer his own reasons in the play, they do not seem to tell the whole story. From his own mouth, however, Iago's reasons are, he tells Roderigo, "hearted," just as Roderigo's "cause" is—Roderigo being driven by his desire for Desdemona. In the opening scene of the play, we discover that Iago has professed his hatred for Othello several times to Roderigo, his supposed confidant; it is questionable, however, as to how far we can believe Iago tells Roderigo the whole truth, as he later seems to have little regard for Roderigo's life.
Iago tells Roderigo that his chief issue with Othello, at this point in time, is that Othello has rejected the suit of "three great ones of the city" who had urged him to make Iago his lieutenant, instead choosing "a great arithmetician," Michael Cassio, who "never set a squadron in the field." Iago believes that he himself is far worthier an officer than Cassio and is furious that he has been passed over for promotion in favor of this lesser soldier. This seems a sound enough reason, and indeed Iago uses Cassio in his scheme to bring down Othello, as if to punish them both. Iago also mentions that he believes Othello "hath leap'd into my seat," or that "twixt my sheets / He hath done my office"—that is, that Othello has slept with Iago's wife, Emilia. However, there seems no justification for this suspicion, and even Iago himself says he doesn't know whether it is true. The idea is simply posited by Iago as if to further justify his actions.
Iago says in Act I, Scene 1 that he hates Othello because Othello has passed him over as a lieutenant. Instead, Othello has chosen Michael Cassio, who, Iago says, has no knowledge of how to be a soldier. Iago describes Cassio as someone "That never set a squadron in the field, Nor the division of a battle knows/ More than a spinster" (I.i.21-23). In other words, Iago believes that Cassio knows less about fighting than a spinster, or old unmarried woman, does.
In addition, Iago suspects that his wife, Emilia, has cheated on him with Othello. Iago says:
"I hate the Moor:/And it is thought abroad, that 'twixt my sheets /He has done my office: I know not if't be true;/ But I, for mere suspicion in that kind,/ Will do as if for surety" (I.3.429-433).
The suspicion that Iago harbors towards Othello is ungrounded, but the very fact that Iago is so jealous of "the Moor," or Othello, poisons him towards Othello. He suspects that the Moor has been in his bed, and, without conclusive proof, Iago decides to act as if his suspicions are true. In general, Iago detests and distrusts all women, and he lets this distrust color his relationship with his wife and with Othello.
There are several reasons given for Iago's hatred of, and intended ruination of Othello. First, he is insulted by Othello's promotion of Cassio; he felt that he deserved the promotion. That is probably the most obvious reason. Second, he is suspicious that Othello is sleeping with his wife, Emilia, although there really is no foundation for this suspicion. Third, there are some hints that Iago really desires Desdemona and is jealous of Iago for his relationship with her.
He is clearly presented as the villain, and as such, can be played simply as evil, but most actors will search for greater depth than this. Most close readers will also look for this greater depth. Many modern productions play up the race issue, painting Iago as a racist who is unhappy with the Moor's marriage to the white Desdemona. There is some textual support for this, in Iago's conversations with Othello when he is trying to convince Othello that he and Desdemona have too many differences to be happy together.