Neither = not either
You can only use them when there are two options.
"I haven't read either of these books." = "I have read neither of these books".
"I haven't read neither..." is incorrect. The reason is more logical than grammatical. Although double negatives are part of Italian grammar, they're not used in English, so a double negative sounds like you're saying (-1)*(-1) = (-1) to an English ear. Double negatives are used in some dialects (namely, African American Vernacular... and I think in New York as well), but otherwise they're never used by native speakers and are wrong.