I some respects he was Achilles' alter ego. His mistake came in thinking he could perform feats that Achilles could. As it was, he performed brilliantly. The counterpart of Achilles. The killer of his friend.
Death is an accepted state of life, that is the end.
The concept of eternal life that is the continuity of after life is borrowed from the eastern religions and developed to its full by christianity. Asterix, Pathroclus became Achilles. And his death announced that of Achilles. Achilles versus Hector is fighting versus the possibility of his own no-existence. It seems significant to me that in his withdrawal from battle, Achilles starts to question this.