Summary: "Live and die by the sword."
It wasn't the first time Achilles caught Patrocolus looking at him. At first it had been with curiosity and awe. As Patroclus grew older, his glances changed, lingered on different parts of Achilles. Rarely was that ever on his eyes.
Achilles has never really looked Patrocolus, or anyone else, in the eyes. He is above them, even those he loves, and he's never forgotten it. It doesn't mean he won't watch them, or that he doesn't notice every move they make. He is the finest warrior who ever lived; he has to be alert. Besides...there is much about Patrocolus to enjoy watching.
Briseis is another matter. She changes things...everything. She revives fascination that he lost when he was a child. Achilles sees her and remembers what the old men in Athens would say: that you learn from men, but you love women. He doesn't know if he loves her, but he's intoxicated by her presence. That's good enough.
When Briseis dies, a part of Patrocolus seems reborn. When Patrocolus dies, the rest of Achilles does, too. The arrow was mercy.