In Richmond Lattimore's translation, the following lines (9.410-416) are a part of Achilles' first speech in Book Nine:
For my mother Thetis the goddess of the silver feet tells me I carry two sorts of destiny toward the day of my death. Either, if I stay here and fight beside the city of the Trojans, my return home is gone, but my glory shall be everlasting; but if I return home to the beloved land of my fathers, the excellence of my glory is gone, but there will be a long life left for me, and my end in death will not come to me quickly.
From this, it is obvious that Achilles knew his life would be shorter if he stayed fighting against the Trojans, and that he didn't know exactly how or when his death would happen. So his pursuit and later duel with Hector might be even more serious to Achilles, that could be suspicious of his own death as a result of the duel. Is this a valid interpretation, or are there any other passages where it is clear that Achilles does not suspect he might die, or even that he knew the duel's result beforehand?