Where did the "doctor" learn his skill?

Chillingworth actually only pretends to be a doctor. He uses his knowledge from past studies in alchemy and things he learned while living among the natives.

"‘My old studies in alchemy,’ observed he, ‘and my sojourn, for above a year past, among a people well versed in the kindly properties of simples, have made a better physician of me than many that claim the medical degree."