Well, the facts of the matter are pretty straightforward. Ahab, the captain of the Pequod, had his leg bitten off by a white whale named Moby Dick, and ever since he has been on a quest for revenge. Of course, things are not so simple. Critics have pointed out that Ahab has much in common with tragic figures from Shakespeare, especially King Lear; similarities have been drawn to Narcissus, or Prometheus, or Satan (from Paradise Lost), or Oedipus. I think, aside from these literary comparisions, the best way to understand Ahab is that he is someone obsessed, not with a whale, but with a particular view of reality.

Ahab's curse is that he believes he understands the secret nature of existence, the bonds of which he is determined to escape. As he puts it in "The Quarterdeck" chapter:

All visible objects, man, are but as pasteboard masks. But in each event—in the living act, the undoubted deed—there, some unknown but still reasoning thing puts forth the mouldings of its features from behind the unreasoning mask. If man will strike, strike through the mask! How can the prisoner reach outside except by thrusting through the wall? To me, the white whale is that wall, shoved near to me. Sometimes I think there's naught beyond. But 'tis enough. He tasks me; he heaps me; I see in him outrageous strength, with an inscrutable malice sinewing it. That inscrutable thing is chiefly what I hate; and be the white whale agent, or be the white whale principal, I will wreak that hate upon him. Talk not to me of blasphemy, man; I'd strike the sun if it insulted me.

Ahab's hatred of "that inscrutable thing" defines him as a character. He has titanic ambition -- he feels he must challenge God. "All visible objects" are false representations of some higher truth -- so in a sense, Ahab sees mortal existence as a lie, a horrible trick played on him by "some unknown but still reasoning thing." His hatred and thirst for vengeance is all consuming. In Chapter 37, he says: “The prophecy was that I should be dismembered; and—Aye! I lost this leg. I now prophesy that I will dismember my dismemberer. Now, then, be the prophet and the fulfiller one. That's more than ye, ye great gods, ever were.” He feels that in killing the whale, he can (figuratively, I guess) become greater than the gods.

It's not that Ahab is without his humanity; he is married, and in Chapter 128 we learn that he has a young son. But Ahab, as a character, is blessed (or cursed) with the gift of insight; his ability to feel anger and vengeance is so developed, so powerful, it overwhelms every other aspect of his personality.

Ahab is characterized as a larger-than-life character, one who is obsessed to madness with finding the "white whale" that struck out at him.  This obsession consumes his thoughts and clouds his judgment at times.  He is a tough, weathered man who is an excellent seaman, but his obsession is his downfall.  It ends up destroying him.  For more about Ahab, see the following link:

http://www.enotes.com/mobydick/characters

Describe Captain Ahab in Moby Dick.

Ahab is set apart from all others by his personal experiences and singular focus on killing his adversary, the whale.  In this light, Ahab is driven by demons that no one else can understand or grasp.  Being abandoned at an early age and convinced that the whale attacked him out of malice, Ahab is unreachable in his desires for revenge.  Ahab is depicted as almost Shakespearean in magnitude, in that he is the victim of tragedy that he himself helped to cause.  Ahab has not understood the need to be able to accept real conditions of consciousness and work through them.  Rather, he is driven to overcome these elements.  He is animated by appropriating the world in accordance to his own subjectivity, as opposed to working through what is there.  In this light, I think that Ahab is the type of figure that is the victim of tragedy that is caused by his own hand.  Accordingly, the idea of Ahab being set apart or being physically and emotionally distant from all others might be a good starting point in describing him.

In Moby Dick, how does Melville develop Captain Ahab's complex character?

Ahab is developed slowly throughout the book, given that he doesn't even appear physically until around the middle chapters.  He begins as a force of nature, Melville describes him as being indomitable, as being willing to do anything to seek his revenge on the creature.  The reader at first sees this as his great power, though there are hints of the somewhat more broken character of the man.

As the plot progresses and Ahab begins to descend into more of a madness, it becomes clear that as he descends, he is more and more a weak man being driven by an obsession, rather than a powerful man seeking some kind of revenge.  He is controlled rather than controlling, representing somewhat of a reversal of his role.  In actuality this was always the case, but Melville takes his time revealing this.

Discuss the characterization of Ahab in Herman Melville's Moby Dick.

One might wonder how Herman Melville was able to write a story such as Moby Dick so realistically. It is because he spent time on a whaling ship that sailed out of the same port from which Ahab's ship departs. The element of realism in this novel of adventure and obsession is born of personal experience.

On January 3, 1841, he sailed from New Bedford, Massachusetts, on the whaler Acushnet, which was bound for the Pacific Ocean. He was later to comment that his life began that day.

The larger-than-life Captain Ahab is an intimidating persona who eventually loses his reason in his pursuit of a white whale with which he has a personal grudge to settle: the loss of his leg.

When Ishmael (the narrator) considers signing on the whaler, Pequod, at the beginning of the story, Ahab is described as “a grand, ungodly, godlike man.” It is not until Chapter 28 that "Old Thunder" (Ahab) "makes a proper appearance." And little information is provided about the captain prior to this voyage. Ahab was orphaned as an infant—at twelve months old. Ahab, ironically, is a Quaker—Quakers are pacifists or peacemakers. Ahab is anything but peaceable. His name comes from the Old Testament. Ahab is married with a child, though he does not provide their names.

Ahab has lost his leg in a recent whaling encounter, and his artificial leg is made of bone: from the jaw of a whale. Melville writes about Ahab's steely dedication to the task at hand, and an unswerving concentration:

There was an infinity of firmest fortitude, a determinate, unsurrenderable wilfulness, in the fixed and fearless, forward dedication of that glance.

Captain Ahab is presented as a man of dignity, but the reader is surprised at Ahab's assertion that rather than being attacked by an animal based upon its instinct, Ahab personifies the whale as malicious. He is also a man who "takes no prisoners." Ahab seems slightly unhinged when he states that he would attack the sun if it had insulted him.

In light of Ahab's need for revenge, his obsession seems to have a life of its own, and it eats away at him.

Ahab’s obsession is presented as a ravenous monster...

The more his fixation takes hold of him, the less rational he becomes. As the story progress, Ahab is presented as one who has lost control of his mind; he is ...

a possessed man at the mercy of his obsession...

In an almost Shakespearean light, Ahab fights his demons alone; they rule his every move. Along with his departure from reason and sanity, he ultimately seals his fate, and that of his crew. Thinking of no one but himself, his actions cause the death of every member of the Pequod's crew, except for Ishmael.

Ahab is destroyed by the very obsession that drives him. As he attacks Moby Dick at the end of the story, he becomes entangled in the whaling gear he is using to kill the whale, and when the whale submerges, it takes Ahab with him. Ahab is presented as a strong man, with a weak mind that give in to his obsession with an element of nature. Like Icarus, who loses sight of his limitations, Ahab loses sight of the fact that he is merely a man, and no match for the white whale.

Additional Sources:

http://en.wikipedia.org/wiki/Herman_Melville

http://en.wikipedia.org/wiki/Ahab_(Moby-Dick)#Ahab