"Who exactly are you?"

Translation: Kdo vůbec jsi?

[Sasoky10]

can't be ( kdo jsi presne )

[OndraSl]

Literal translation is, as you said, "Kdo přesně jsi?".
Could be used, tho a czech person would generally use "Kdo vůbec jsi?" (who are you actually?).

[ValaCZE]

This translation is also accepted.