Here’s something that should put your mind at ease:
I, George Saoulidis, give you permission to have a black/asian/latinx/whatever Achilles in your story. I also retroactively give you permission to have any possible combination of skin colour/sexuality/hero name from the Greek epics.
How the fuck can I possibly give permission for something like that, you might ask?
That’s my point. The same way that it’s nobody’s business to give you permission to do something like this, nobody can stop you from doing it either.
The Greek epics do not belong to me, you, or any other Greek in the world. They belong to everybody. So much so that the words “public domain” are too puny to describe them. They’re infused into our culture, shaped the very pathways in our brains. They have inspired endless creators for literal millennia (pause and let that sink in for a moment.)
So, do you want to take a popular myth and make it your own? Are you inspired by Odysseus or Achilles or Zeus and want to see yourself in their shoes, and you also happen to not be white and male? Do you want to cosplay as them, write about them, make films and games and VR experiences about them? Do you want to create and have found inspiration in the most influential and badass mythology of all time?
By all means, go for it.
That won’t mean that I’ll like your story. It won’t mean I’ll spend money on your story. If it objectively sucks, I’ll happily tear you a new one. Or worse, I’ll just ignore it. But if the story is good, if your vision is great, if your interpretation is modern and touches the souls of those around you, then go for it.
Nobody’s stopping you. Not me. Certainly not Homer.