The best term for this is self-styled god or self-proclaimed god. Using deify with a reflexive pronoun, like he deified himself, could also work.
Self-styled or self-proclaimed
English has a lot of terms for someone who names himself something: self-styled, self-declared, self-proclaimed, and so on. Many of these have their origins in medieval titles, or styles, so a self-styled king would be someone who is not recognized as king by any country, but refers to himself as king and tries to insist on royal honorifics (your majesty and so on).
However, these terms are generic and can apply to any title or honor—including divinity. A self-styled god or self-proclaimed god would be the same as a self-styled king, but for divinity rather than royalty.
Deify—but not self-deify
To deify is to make something into a god. It could be used to refer to the actor him-or-herself, as in he deified himself.
The construction self-deify is understandable, but awkward and basically unheard of—see this graph of usage of deify vs. self-deify, where deify enjoys some usage, but self-deify is literally not found at all. So self-deify could be English, but not idiomatic English.
A similar word is apotheosis, which means to become a god. Self-apotheosis, though, doesn’t really make much sense, and apotheosis is less well-known anyway. It’s also more positive—the term usually refers to someone who actually does become a god (it’s from Greek; that sort of thing happens in Greek myths). Deify instead refers more to someone who is worshiped as a god.