Odysseus kills Antinous first because he was the suitor's leader, both morally and politically. Antinous was the richest, loudest, most obnoxious, and most ungrateful of the suitors. He abused Odysseus by throwing insults and then later a chair at him. Further, Antinous doesn't suspect a thing, and Odysseus must eventually kill them all anyways.
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Odysseus shot unsuspecting Antinous through the throat with an arrow from his mighty bow, as Antinous was eating an apple.
Antinous never wants to kill Odysseus, because he never realizes Odysseus is still alive. He may have wanted to kill the beggar, whom is Odysseus in disguise, but is prevented from doing so by honour and Telemachus' threats. Odysseus as a beggar begs from Antinous and curses him when Antinous refuses impolitely. He does want to marry Penelope, Odysseus' wife.
The suitors believe Odysseus to have killed Antinous by mistake, and they threaten to kill him as well.
Odysseus uses his large bow and an arrow to kill Antinous. The bow was given to Odysseus from Iphitus, who received it from his father Eurtyus, who received it from his grandfather Apollo.