In Shakespeare's Othello, why does Iago feel that he should have Cassio's job?

Iago is a complex and duplicitous character. That means we can never fully trust what he says, even when he is speaking a soliloquy. Part of his belief that he deserved the promotion is grounded in his own sense of entitlement.

He does, however, give a clear account of why he thinks he is better qualified than Cassio in the first scene of the play. 

The first and most explicit reason why he believes that...

Iago is a complex and duplicitous character. That means we can never fully trust what he says, even when he is speaking a soliloquy. Part of his belief that he deserved the promotion is grounded in his own sense of entitlement.

He does, however, give a clear account of why he thinks he is better qualified than Cassio in the first scene of the play. 

The first and most explicit reason why he believes that he deserves the promotion is that he is senior to Cassio and has been serving longer. Next, Cassio is a Florentine rather than a Venetian. Also, Cassio is a "bookish theoric" rather than someone who has substantial battleground experience. Iago mentions that he has served in far more places than Cassio and had given proof of his skill as a soldier in that service while Cassio is basically untried. 

Finally, Iago believes in the value of an older tradition "where each second / Stood heir to the first" and he does not feel Othello should simply have the right to choose his own lieutenant "by letter and affection."