Is "I'm dead serious" formal?

Dead, as adverb, is informal when used to mean very, such as in "omelets are dead easy to prepare." (As far as I know, dead is used to mean very only in British English.)
When dead is used to mean absolutely, completely, exactly, straight, or directly it is not used only in informal phrases.

Well, the use of a contraction isn't formal, but putting that aside …

Since the intended meaning of the word in that phrase does match the literal definition of dead as an adverb, it must technically be considered formal. However, such usage is certainly secondary to its common interpretation as a noun, and is liable to confuse your audience across cultural and language barriers. In formal writing, I'd avoid it just in case the people judging your work falsely condemn it.

Neither is the phrase offensive, but the primary meaning of dead still lends it a slightly morbid connotation. I'd use it sparingly and in appropriate context. For example, I'd say I'm dead serious in a business meeting, but not you're dead right. The latter statement clashes with the adverb's connotation, which makes the word stand out rather than just effectively carry your thoughts.