When the Greeks managed to conquer the city of Troy, using the trick of the Wooden Horse, they engaged in wholesale slaughter. This vase shows one of the most outrageous scenes of that story: Neoptolemus (also called Pyrrhus), the son of Achilles, slaughtered Priam, the aged king of Troy, and he did so upon a sacred altar. What is he holding in his hand? That is the corpse of Hector's young child, Astyanax, whom Neoptolemus also killed, and now he is using the dead boy to club to death the boy's grandfather. Blood-thirsty indeed. Hence the proverb, Heroum filii noxae, "The sons of heroes are criminal offenses." Neoptolemus is a chief instance of this pessimistic principle.
You can read more about Neoptolemus at Wikipedia. You can read details of the story in Vergil's Aeneid, Book 2.