Achilles just killed memnon because he had killed Nestor Achilles's friend. Then challenged any other Greek to fight him so Ajax challenged him but Achilles wanted to challenge him, himself so he did and ended up winning the duel.
After killing Memnon Achilles shouted to the Greeks and raised his hand in Triumph. That was when Paris cast an arrow in his bow and waited for the right shot and fired. It ended up hitting him in his heel the only mortal place that Achilles had and killed him.
Hermes was sent to tell Agamemnon to listen to the next person who comes up with a way to get inside the gates of Troy. So he did and he listened to Panopeus who said that they should make and fill a wooden horse with their best men and then all send their ships away.
Most of the Greek army left on their ships and set out for home and the rest of the warriors stayed and hid in the belly of the horse. While scouting and looting he battle field they came across a lone soldier and he told them that the horse was meant to be a sacrafice to the gods but they never used it and left it for the Trojans.
The Trojans thought nothing of it and brought it inside the walls of Troy and left it then partied because of their victory against the greeks. Then they just left it in the middle of the city overnight.
The Greeks broke out of the horse and ended up looting, killing, and enslaving the city and set fire to it and ended the war.