“…and then especially, where he speaks of Priam’s slaughter”
This tale is told in both Virgil’s Aeneid and Marlowe’s Dido.
The First Player tells the story of the death of Priam, caused by Pyrrhus, the son of Achilles. Pyrrhus fights in the Trojan War because his father fought in the war and was killed. Pyrrhus kills Priam mercilessly on an alter after the fall of Troy.
Hecuba, Priam’s wife, stands by helplessly while Priam is murdered. Her grief is unbearably profound. How has Gertrude reacted to the death of Old Hamlet?
The story could be relating Claudius to Pyrrhus and Old Hamlet to Priam. The story also reminds Hamlet of the revenge he is seeking for his own father.