Who is the first person that Odysseus kills and why does he kill him first? After this first man is killed, what bargain does Eurymachus offer to Odysseus?
First, we should note that Odysseus was a hero of the Trojan war, and thus has a long and bloody history of killing people both during the war and on his journey home.
After Penelope's stratagem with unweaving the shroud has been discovered, she sets a second condition, that she will marry the man who can draw the string of Odysseus's bow and shoot an arrow through twelve ax handles. Odysseus, disguised as a beggar, conspires with the servants to lock the doors to the hall and is granted a chance to try the bow. He shoots an arrow with almost contemptuous ease through the ax handles. Next, he shoots Antinoös, leader of the suitors, in the throat.
Eurymachus casts blame on the dead Antinoös, and offers the following as restitution:
Each one of us shall pay you a fine worth twenty oxen, and we will keep on giving you gold and bronze till your heart is softened.
Odysseus first kills Antinous by shooting an arrow at him while he appears to be drinking from a cup. The buttshaft of the arrow is sticking out of his neck when all is said and done. This is the man who was most after his wife and sort of was the ring leader of the men vying for Penelope's hand in marriage.
Eurymachus argues this very point (Antinous being the ringleader) and argues that the men will pay back all they have taken and then some. Odysseus doesn't even consider the offer and proceeds to take out all of the men until they are all dead.
The first person that Odysseus kills is Antinous. We are not told exactly why Odysseus kills him first. Presumably, it is because he is one of the two leaders of the suitors and Odysseus wants him out of the way.
After Odysseus kills Antinous, Eurymachus tries to make a deal with him. Eurymachus tells him that the suitors will make amends for what they have done. They will pay Odysseus back for all of the food and drink that they have consumed. They will also each pay a fine to Odysseus. Odysseus, of course, refuses.