Achilles and Hector

After the war, Hector’s son, Astyanax, was thrown off a building

It was said that Achilles’s immortal horses could speak

After the fall of Troy, Andromache left with Neoptolemus

Achilles Heel, Achilles and Hector both in Troilus and Cressida, Achilles in Ender's Shadow

http://www.sparknotes.com/lit/iliad/summary.html

http://www.history.com/topics/ancient-history/achilles

http://ancienthistory.about.com/cs/grecoromanmyth1/g/hector.htm

http://www.britannica.com/EBchecked/topic/3581/Achilles

http://ansible.wikia.com/wiki/Achilles

Achilles was the most powerful warrior in the Iliad and the army of Agamemnon, proud and headstrong. He took offense to things easily, and has wrath like a god.

Near the end of the Trojan War, Patroclus, Achilles' beloved comrade, goes out into battle wearing Achilles' armor. Recognizing the ensignia, Hector fights and kills Patroclus. When news of this reaches Achilles's ears, he is enraged, causing him to go on a rampage killing every Trojan in his path, and ending with Hector. After killing Hector, Achilles ties him to a chariot in order to disgrace his corpse, dragging him around Troy multiple times, and Achilles finally relents to the Trojans, allowing them the body to give Hector a proper burial.

Achilles: 

Parents - Peleus and Thetis 

Offspring -  Neoptolemus

Mating Partner - Deidamia, princess

Hector:

Parents - Hecuba and Priam 

Offspring - Astyanax

Mating Partner - Andromache

Epithets of Achilles:

Swift-Footed Achilles 

Leader of Men

Shepherd of the People 

Epithets of Hector:

Tamer of Horses

Priam's Son of the Glinting Helmet