How does Othello deal with his role as a leader?
Not very well.
You have to look at what a leader is expected to do. A leader should inspire confidence, and Othello clearly does this because the Duke and the others insist on sending him. They have faith in his abilities as a warrior, and as a fighter, we have good evidence that Othello excels.
However, a leader should also be able to arrange his resources to get the most out of them. Othello first promotes and then discharges Cassio, so he does not make good use of the man's talents.
A leader should understand the men under him and head off potential problems. Othello worked with Iago, and if Iago has heard rumors about Othello and Emilia, you would think Othello would have heard the same rumors. However, Othello doesn't ever catch onto the fact that Iago is furious and jealous.
A leader should have faith in his own judgment and worth, and this is where Othello fails the most. He sees himself as being beneath the others, as having no soft parts of speech, as being ugly and unworthy. Considering that he has endured slavery and discrimination, this attitude isn't surprising, but it does undermine him as a leader. This flaw allows Iago to weave his plot and undo all of Othello's good attributes.