I studied ancient sexuality in college and I have been waiting for this question, thank you.

The perception that Achilles and Patroclus were lovers was propagated by classical-age Athenians because that was the model for such a close male friendship that the Athenians had at that time. However, they were imposing classical age norms on a text that was written in the Greek dark ages (8th century BCE), and was set in the Mycenaean period (around 1200 BC).

We have far, far less of a clear picture of dark age and Mycenaean sexuality than we do of the classical age so it becomes very difficult to conclusively say whether or not the two were lovers based on cultural analysis. Textual analysis becomes bungled in translation, but I tend to fall on the side of the two being lovers.

Regarding their sexual orientation, the work of scholars such as Michel Foucault and Kirk Ormand suggests that prior to about the 16th century CE, sexuality had not been constructed as an identity. I like to explain it like this: sexual preference in the ancient world was like coke and Pepsi today; most people have a preference, but enjoy both, and no one categorizes by their favorite. In fact, those who do are seen as sort of weird and fanatic.

So were they homoerotic? I think so. Were they gay? No. “Gay” is an identity which was constructed within the last 100 years and does not at all fit with the behavior of two ancient warrior-princes who themselves were incredibly privileged, (unlike the modern queer community) also liked women, (which would make them bi if anything, but that doesn’t hold for the same reason) and would have been offended if you had suggested that their behavior was in any way deviant, (since the hallmark of the modern LGBTQIA+ community is deviation from heteronormative patriarchy.)