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Priam's greatest fault is being a pushover for his son Paris. Even though everyone thinks Paris should just give Helen back to the Achaians, Priam lets his son have his own way, thereby condemning his city to eventual destruction at the hands of the Achaians. That said, modern readers will no doubt sympathize with his deep love for his children, which makes him constantly worry about their safety (e.g., when he can't stay to watch the duel between Paris and Menelaos in Book 4, or when he and his wife, Hekabe, implore Hektor to come inside and not face down Achilleus in Book 22.)
Priam is also remarkable for the courage he displays in Book 24, when, with divine guidance, he goes unarmed to find Achilleus and ask for the return of Hektor's body. Oh yeah, and one other thing: even though he's an old man now, back in the day, he had something like a thousand concubines, and practically half the Trojan army is made up of his sons.