In Shakespeare's Othello, Iago is really at the center of what transpires, and it is based on his hatred for Othello.
The reader learns that Iago is Othello's "ensign" or "ancient." Iago thinks he will be promoted, but Othello promotes Cassius instead. Iago's rage know no bounds, and he begins to plot in the tradition of some of Shakespeare's greatest villains (e.g., Cassius in Julius Caesar) to bring about Othello's downfall.
In Act One, scene one, Iago speaks about his anger for Othello—ignoring Iago even after at least three men of importance in the city have recommended him. Iago notes that Othello avoided these recommendations on Iago's behalf. Iago states that he knows he is a valuable asset to Othello, but instead Othello chose Cassio—a Florentine, for Heaven's sake!)—who has never commanded a regiment and knows nothing of battle, unless he has read it in a book.
Three great ones of the city,
In personal suit to make me his lieutenant,
Offcapp'd to him; and, by the faith of man,
I know my price, I am worth no worse a place.
But he, as loving his own pride and purposes,
Evades them with a bombast circumstance
Horribly stuff'd with epithets of war,
And, in conclusion,
Nonsuits my mediators; for, “Certes,” says he,
“I have already chose my officer.”
And what was he?
Forsooth, a great arithmetician,
One Michael Cassio, a Florentine
That never set a squadron in the field,
Nor the division of a battle knows
More than a spinster; unless the bookish theoric... (9-20, 22-24)
It is Iago's plotting that will bring Othello down, as Iago plays on his commander's passionate jealousy.
Regarding Cassio and Iago, Iago hates Cassio because he has been promoted over him. Cassio has no issue with Iago—however, Iago will use Cassio to make Othello jealous. In Act Two, scene three, Cassio is Iago's pawn when Iago gets Cassio drunk, hoping to get him in trouble.
Come, come, you're drunk.
Drunk! They fight.... (146-147)
As Cassio and Montano return, Othello has come upon the scene, demanding an explanation.
How comes it, Michael, you are thus forgot?
CASSIO:I pray you, pardon me; I cannot speak... (179-180)
Iago speaks instead, convincing all that he loves Cassio and would never harm him, but must speak the truth...one that serves his purposes—but does not explain what really happened. Othello says:
I know, Iago,
Thy honesty and love doth mince this matter,
Making it light to Cassio. Cassio, I love thee;
But never more be officer of mine. (239-242)
Upon discovering this "truth," Othello dismisses Cassio for "behavior unbecoming of an officer." But this is not enough for Iago. Iago convinces Cassio to go to Othello's new wife, Desdemona, to beg for his position back, asking her to intercede for him. She says:
O, that's an honest fellow. Do not doubt, Cassio,
But I will have my lord and you again
As friendly as you were. (III.iii.5-7)
Iago notes how chivalrous Cassio is with Desdemona—they have an innocent affection for each other, but Iago twists this truth. When Emilia finds Desdemona's handkerchief, Iago takes it and tells Othello that Desdemona wiped Cassio's face with it—that's why Cassio has it. Another lie. Iago then has conversation with Cassio (in the presence of a hidden Othello) that speaks of Cassio's relationship with Bianca, but Iago has told Othello it is about Desdemona.
Othello ultimately kills Desdemona and himself. Iago hates Othello, and uses Cassio to get his revenge.