Iago is an interesting villain. Whereas many of Shakespeare's villains—like Macbeth, Lady Macbeth, Angelo (in Measure for Measure), Tamora (in Titus Andronicus), and Richard III—generally take care of business themselves or directly order someone else to do it, it's not until the last act of Othello that Iago does anything to get his own hands dirty. Even then, Iago gets personally involved only because his plan against Othello goes slightly awry, and he has to step in to sort things out by wounding Cassio and murdering Roderigo in act 5, scene 1. Later, in act 5, scene 2, Iago kills his own wife, Emelia, for revealing his plan to destroy Othello and Desdemona, but this occurs only after his plan has already succeeded. Until then, Iago is content to manipulate others into doing his dirty work.
Iago is a master of insinuation and deception, and he uses those skills to take advantage of other characters' weaknesses and insecurities. Othello is a trusting soul, and Iago insinuates himself into Othello's trust— "I know thou'rt full of love and honesty" (3.3.134), "My friend... honest, honest Iago" (5.2.185). Iago also recognizes that Othello has a dangerously jealous nature, and he insinuates to Othello that Desdemona is unfaithful to him after Othello sees Cassio talking with her and quickly leave the scene when Othello enters.
IAGO. Ha! I like not that.
OTHELLO. What dost thou say?
IAGO. Nothing, my lord; or if I know not what.
OTHELLO. Was not that Cassio parted from my wife?
IAGO: Cassio, my lord! No, sure, I cannot think it,
That he would steal away so guiltylike,
Seeing you coming. (3.3.37-43)
In act 4, scene 1, Iago deceives Othello with a staged conversation with Cassio that leads Othello to believe that Cassio is talking about Desdemona, when in fact Iago and Cassio are talking about Bianca, a prostitute who's in love with Cassio.
Unlike other Shakespeare villains who have clearly defined motives and objectives for their villainy (like wanting to be King of England or of Scotland, for example), Iago seems to lack a convincing motivation for wanting to destroy Othello and Desdemona.
IAGO. I hate the Moor. (1.3.379)
There's a racist motive involved, but it's difficult to know if it's Iago who's racist—he constantly refers to Othello as "the Moor"—or if Iago is simply playing on the prejudices of other characters like Desdemona's father, Brabantio.
IAGO. 'Zounds, sir, you're robb'd! For shame, put on your
Your heart is burst, you have lost half your soul;
Even now, now, very now, an old black ram
Is tupping your white ewe. Arise, arise!
Awake the snorting citizens with the bell,
Or else the devil will make a grandsire of you.
Arise, I say!
...[Y]ou'll have your
daughter covered with a Barbary horse; you'll have your
nephews neigh to you; you'll have coursers for cousins,
and gennets for germans. (1.1.91-97, 126-127)
Iago was passed over by Othello for the position of lieutenant, and he resents Othello for promoting Cassio instead of him. (1.1.8-33) This has nothing whatsoever to do with Desdemona and seems a rather insubstantial motivation for everything that Iago perpetrates against Othello in the play. Iago also thinks that his wife, Emilia, might have been unfaithful to him with Othello, but Iago doesn't seem particularly upset by that possibility and dismisses it.
IAGO. ...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. (1.3.397-400)
Later in the play, Iago uses Emilia's supposed infidelity to rationalize his lust for Desdemona and his desire to destroy Desdemona along with Othello.
IAGO. Now, I do love her too,
Not out of absolute lust, though peradventure
I stand accountant for as great a sin,
But partly led to diet my revenge,
For that I do suspect the lusty Moor
Hath leap'd into my seat; ...
And nothing can or shall content my soul
Till I am even'd with him, wife for wife;
Or failing so, yet that I put the Moor
At least into a jealousy so strong
That judgement cannot cure. (2.1.299-310)
Nevertheless, Iago has no deep or profound reasons for his all-consuming hatred of Othello. The few reasons that Iago confides to the audience and to other characters serve only to justify his behavior rather than to support any substantial motivations for his actions.
Quite simply, Iago doesn't need any reason for his malicious, destructive behavior other than his desire to take revenge on anyone who crosses him or gives him even the slightest provocation.