Why does Iago think he deserves the lieutenancy?
Iago's reasons are partly based on reason and partly based on his own character. He resents Othello even more than Cassio, and is an exemplar of what now would be called "white privilege." Although Iago sometimes advances rational motives for his malignancy, often he seems simply to regard anything not honoring or benefiting himself as a direct injury and reacts with a combination of spite and self-aggrandizement.
In the case of the promotion, he argues that he has more actual practical experience as a soldier than Cassio, whom he describes as follows:
One Michael Cassio, a Florentine
(A fellow almost damn'd in a fair wife),
That never set a squadron in the field
But he, sir, had th' election ... (1.1.20-27)
He feels that as a Venetian, he had more right to the promotion than an inexperienced Florentine.
Cassio is younger (and handsomer) than Iago and Iago feels that as an older and more experienced soldier he should be promoted over a younger one. He is also jealous of Cassio's attractiveness to women and Othello's trust in him. He also, though, is jealous of Othello, feeling that it is unfair that an older, black man manages to woo and win the lovely young Desdemona. Basically, Iago's motives and beliefs are not entirely rational.
I would say there are two reasons.
First, he thinks he deserves it because of the recommendations of the three important men. They personally asked Othello to take Iago as lieutenant.
Second, Iago says that he has way more experience in battle than Cassio does. Cassio, he says, has never been in battle.
So Iago has more experience and he has the backing of important people and so he thinks he deserves the higher post more than Cassio does.
Iago feels that he is far better placed to be lieutenant than the 'mere arithmetician' Cassio. Iago has been at Othello's side through many battles and, as illustrated by Othello when he asked Iago to escort his new wife Desdemona to Cyprus, is both trusted and relied upon by Othello.