Iago addresses just this question in Act I, scene i. He explains to Roderigo (and the audience) the grievence that he has with Othello's preferment of Cassio over him. When Roderigo says, "I would not follow [Othello] then," Iago makes it clear that he follow no one. He answers, "I follow him but to serve my turn upon him," and "In following him, I follow but myself."
This confession sets up the false behaviour Iago will exhibit throughout the rest of the play. He, on numerous occasions, finds just the way to behave that will prey upon his victim's weakness and make Iago, at the same time, appear to be his or her best friend. Here are some of the ways he makes himself favorable to others:
- He convinces Roderigo that he has a chance to get Desdemona's love, and that he, Iago, is his ally.
- He convinces Cassio both to drink too much, and, once that leads to his dismissal, he convinces him to ask Desdemona for help. All this in the guise of being Cassio's buddy.
- He convinces Othello, mostly by swearing that it is NOT true, that Othello's wife and Cassio are having an affair.
In Act I, scene i, not only does he explain the situation that has led to his desire to "serve his turn" upon Othello, but he states right up front that he will hide his true feelings and opinions. He says:
Heaven is my judge, not I, for love and duty,
But seeming so, for my peculiar end.
. . .I am not what I am.
Iago is able to make himself look favorable in others' eyes by "seeming" to be full of "love and duty," while beneath, he hides his true intentions.
For more on Iago's "seeming," please follow the links below.
I think that one of the most intense ways in which Iago makes himself look favorable in other's eyes is because he is able to inspire trust within others. Iago's history of being Othello's ancient is something that he uses to his advantage. In concocting his own manipulation of others in the play, the critical element is that he is able to inspire others' trust, causing him to be seen as favorable in the eyes of others. He understands what fundamentally motivates other people and in this, he is able to plot or scheme so that he is able to gain their trust and serve as an adviser. This appears consistent with his advising capacity to Othello on the battlefield, so the impression people have of him is concurrent with what he has already shown. Through being able to gain others' trust as part of his larger design, Iago is able to make himself look favorable to others.