Well, Herakles appears first in Homer, and his apotheosis is already part of the story, so I guess strictly speaking, he's Greek. This is the incident where Odysseus encounters the shade of Herakles in the underworld, and notes that Herakles himself is on Olympus with the gods. The writer Lucian later will make fun of how ridiculous this is in his short form Dialogs of the Dead.

However Herakles is kind of a generic culture hero in a lot of ways. Romans might have adopted the name from the Greeks, but traits of the culture hero occur for instance in the Sumerian hero Gilgamesh, who wears a lion skin, carries a club, and confronts mortality, as well as performing tasks like making the cedar forest safe for civilization.

Hercules (or Heracles) is mentioned the Iliad. To take the first relevant quote from there:

Tlepolemus, son of Hercules, a man both brave and large of stature, brought nine ships of lordly warriors from Rhodes.

That was probably written in the 8th century BC, based on an earlier oral version. This is around the time Rome was founded. I expect there is earlier evidence of him in Greek culture.

That's not to say that all the legends associated with him are Greek in origin. Heracles is interesting because, according to Arrian's history of Alexander, there were Heracles legends as far afield as Afghanistan or Pakistan. Many of these, presumably, were indigenous legends that the Greeks identified as Heracles.

I don't have a source to support it, but I'd place money on the fact that Rome's Hercules has its origins in Greece, rather than being an independently created legend that was merged with the Greek hero. However I'm not sure how many of the best known tales about Hercules are Greek in origin and how many Roman.