In the context of "How deep\ly should I study something."
In that context, it's deeply.
Deep is an adjective, so can only be used to describe the quality of a noun:
Deeply is an adverb, so can only be used to modify a verb:
Deep is often used instead of deeply, to the extent that it is largely accepted without comment. However, I've never heard it the other way around. "What a deeply question" is just plain wrong.