Disassembling a faucet to replace its cartridge
You need to get a small mirror and bright flashlight and reflect the light into that hole so you can see what you're dealing with. Just about every screw in those handles needs an Allen wrench to remove it. It could be SAE or metric.
Once you get that handle off, you have to loosen the chrome collar at the seam you can see in your photo. A strap wrench or leather belt (Thanks Ed) works so you don't damage the surface.
These faucets are tricky so don't get too discouraged if things don't go right the first time.