How does Shakespeare present Iago across the play?

Like many of Shakespeare's major characters, Iago is both an individual and a larger-than-life personification of one or more abstract qualities, in his case power-lust and envy. Throughout the play there are seeming inconsistencies in Iago's behavior, which can be understood only in the context of this type of duality in Shakespeare's characterizations.
In the Italian romance from which Shakespeare derived the plot of Othello, Iago's principal motivation should be his own love for Othello's wife Desdemona. In Shakespeare, we don't get any sense of this, although it's possible that desire for Desdemona is an unspoken or unconscious subtext of Iago's envy. He is inflamed more by the fact that he has been passed over for a promotion, but this alone doesn't explain the virulence of his hatred and the plot he sets in motion.
Throughout the play the reader, or audience, is impressed by a mystery at the heart of Iago's actions. It would be too easy to say he represents "pure evil," though there's a grain of truth in this. Significantly, when Iago is taken prisoner in the final scene, his only explanation is to say, "Demand me nothing. What you know, you know. / From this time forth I never will speak word." Shakespeare leaves Iago's deepest motives an open question, just as, in the world overall, irrational and unexplained impulses are at the root of crime and tragedy. That Othello attempts to kill Iago but only succeeds in wounding him is another sign that man can't destroy evil, that it exists as a perpetual force that can neither be eliminated nor controlled.
Elsewhere in the play, however, Iago shows more obviously human, or even vulnerable, traits, despite as reprehensible as he is. He makes crude racial remarks about Othello. He also has a childish streak and takes pleasure in obscene jokes, describing Othello's and Desdemona's love-making by saying to Brabantio "Your daughter and the Moor are now making the beast with two backs." His ironic toast, "Here's to happiness between their sheets" is another sign of Iago's obsession with sexuality. Yet, as with all Shakespeare's major characters, he speaks elsewhere with an eloquence that is at odds with his crudeness and scheming. His most famous quote is probably "For when my outward action doth demonstrate / The native act and figure of my heart / In complement extern, 'tis not long after / But I will wear my heart upon my sleeve for daws to peck at." The verses are sometimes interpreted as indicating that Iago does, in fact, wish people to see into his heart. In context, it's clear he's saying the opposite, since he closes with "I am not what I am," but he still reveals a vulnerability at odds with his viciousness and with the demonic plan he sets in motion.
These "contradictions" in Iago, if that's what they are, are typical of Shakespeare's characterizations. Hostile critics such as Leo Tolstoy, who late in life wrote a long essay in which he enumerated Shakespeare's faults as a writer, saw them as defects. It may be partly true that Shakespeare regarded poetic eloquence as an end in itself and did not care if the sentiments expressed matched the person who spoke them. But the paradoxes within his characters indicate how deeply Shakespeare understood not just the inconsistencies but the irrationality that is at the root of human behavior. Iago's actions do not stem from reason but from a primal hatred that is a mystery to the ordinary law-abiding person who has no desire to harm others. The portrayal of Iago, with all his contradictions, is perhaps the most compelling and memorable element in Othello.