Captain Ahab seeks revenge against the white whale that he perceives as a supernatural creature of malice.

At one point in his narrative, Ishmael describes Captain Ahab:

"While his live leg made lively echoes on the deck, every stroke of his dead limb sounded like a coffin-trap. On life and death this old man walked." (Ch. 51)

Ahab seeks the white whale with a monomaniacal desire for revenge. When the first mate, Starbuck, repudiates the idea of chasing Moby Dick, exclaiming,

"Vengeance on a dumb brute...that simply smote thee from blindest instinct! Madness! To be enraged with a dumb thing, Captain Ahab, seems blasphemous." (Ch. 36)

Ahab answers Starbuck's charges, saying that everything that is visible is but "as pasteboard masks." It is behind these masks that lies the unknown "but still reasoning thing[s]." Therefore, man must strike through these masks if he is to know what lies behind them.

"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." (Ch.36)

Ahab perceives the white whale as a mysterious force of evil that he desires to conquer. He wants to "break through" to the supernatural force that lies behind the physical being of the whale and destroy this if he can.

Simply put, Ahab seeks revenge because the whale caused him to lose his leg.

Ahab does not see the whale as merely "a dumb brute...that...smote (him) from blindest instinct" Chapter 36).  He sees in him rather "outrageous strength, with an inscrutable malice...(that) insulted (him)" (Chapter 36).  He is obsessed, compelled to seek vengeance by an "innermost necessit(y) in (his) being" (Chapter 36), "damned in the midst of paradise" (Chapter 37), by "madness maddened" (Chapter 37).  His heart is consumed by hatred and a deep-seated need to seek revenge, and he vows to pursue "that accursed white whale that...made a poor pegging lubber of (him)...round Good Hope, and round the Horn...and round perdition's flames before (he) give(s) him up" (Chapter 36).

Ahab seeks revenge for two reasons.

One he has been hunting Moby Dick for years and has not been able to kill him. He is frustrated by his inability to catch and kill  Moby Dick.  He personifies the whale into his mortal enemy.

And two, he lost his leg to Moby Dick so he is even more determined to kill him.