Macduff wants to kill Macbeth because Macbeth killed his entire household, and because Macbeth is driving their country into the ground.
Everyone wants the bloodthirsty Macbeth gone, but Macduff has a very personal reason. Macbeth had his wife, son, and household murdered.
All my pretty ones?
Did you say all? O hell-kite! All?
What, all my pretty chickens and their dam
At one fell swoop?(255) (Act 4, Scene 3, enotes pdf p. 74)
Malcolm tries to convince Macduff to channel his grief into revenge.
Let's make us medicines of our great revenge,(250)
To cure this deadly grief. (Act 4, Scene 3, enotes pdf p. 74)
Of course, Macduff does not need much convincing. He needs to avenge to terrible, unnecessary slaughter of his family. Revenge works best when you do it yourself. Macduff also feels responsible.
Macduff is also "the first character to suggest his suspicion regarding Macbeth's ascension to the throne" (character analysis).
They were all struck for thee! Naught that I am,
Not for their own demerits, but for mine,
Fell slaughter on their souls. Heaven rest them now! (Act 4, Scene 3, enotes pdf p. 74)
Macduff realizes that his family was killed because of his relationship with Macbeth, so he feels the need to personally avenge them.
Macduff is a loyal countryman and hates seeing Macbeth rule Scotland like a tyrant. In Act Three, Scene 4, Macduff has a conversation with Malcolm where he expresses his displeasure that Macbeth is ruining their country. Macduff tells Malcolm that each day widows howl and orphans cry because of Macbeth's tyrannical reign. In addition to wanting to rid his country of Macbeth's tyranny, Macduff has a personal vendetta against Macbeth. Ross informs Macduff that Macbeth has brutally murdered his wife, children, and servants. Macduff then vows to kill Macbeth for slaughtering his family. He tells Malcolm,
"Front to front bring thou this fiend of Scotland and myself. Within my sword’s length set him; if he ’scape, heaven forgive him too" (Shakespeare, 4.3.239-243).
In Act Five, Scene 8, Macduff gets his revenge on Macbeth when he meets him face to face during the final battle. Macduff ends up killing Macbeth and decapitating him at the end of the play.