In Agamemnon, what arguments does Aegisthus use to prove that the gods were in favor of Agamemnon's being murdered, and that he was justly killed?
Aegisthus sees Agamemnon's murder as justice because of what Agamemnon's father, Atreus, did to his own father. Atreus fed Aegisthus's father a meal of his own children, feigning friendship. When Thyestes realised what he had consumed, he cursed Atreus and his descendants. Aegisthus sees Agamemnon's murder as the fulfilling of this curse and the way that Atreus was so cruel to Thyestes and his family. Note what Aegisthus says about the justice in Agamemnon's death:
This is the reason that you see this man fallen here. I am he who planned this murder and with justice. For together with my hapless father he drove me out, me his third child, as yet a baby in swaddling-clothes. But grown to manhood, justice has brought me back again.
Aegisthus clearly repeats the word "justice" to reinforce and strengthen his claim that the murder of Agamemnon was just and deserved. The way that he claims responsibility, or ownership for this murder, shows how strongly he believes that this murder was a good thing, and that it was enacting justice, and that there is a sense in which it is only fitting that Aegisthus, who has suffered so much as a result of Agamemnon and his family, should be the one to organise his death and return to the kingdom from which he was exiled.