The most famous description of Hector's death occurs in Book 22 of Homer's Iliad. In this book, Achilles, having reconciled his quarrel with Agamemnon, returns to the fighting and is intent on killing Hector, who killed Patroclus, who was Achilles' best friend.
Early in Iliad 22, Priam, watching Achilles close in on Hector, pleads with his son not to face Achilles alone. After Achilles kills Hector and begins dragging Hector's body around Troy behind his chariot, Priam "gave a pitiful groan" (22.408; Ian Johnston's translation), expressed "frantic grief" (22.413; Ian Johnston's translation), and began rolling about in the dirt and begging people to leave him alone in his grief.
In Iliad 24, Priam manages, with the help of the gods, to sneak into the Greek camp, make his way to Achilles' tent, and persuade Achilles to ransom back to him the body of Hector.