Learn Croatian: Nemojte kriviti mene!

mene is the full (emphatic) form of the pronoun ja (I) in the accusative case. We use the full form here to emphasize the pronoun: "Don't blame me!" (implying you should blame someone else instead).

If I didn't want to add emphasis, could I just swap mene for me at the end of the sentence?

No, because the word order would have to change. Short pronouns like me are "clitics," meaning they naturally need to be in the second position of the sentence. Without the strong emphasis, you would say Nemojte me kriviti!

Why do we use Nemojte instead of just Ne to say "Don't"?

To make a negative command (telling someone not to do something), Croatian usually uses the special helper verb nemoj (for a singular, informal "you") or nemojte (for a plural or formal "you"). You follow it directly with the infinitive form of the verb, which is kriviti (to blame) here.

Why is the pronoun in the accusative case?

The verb kriviti (to blame) requires a direct object, because you are directing the action of blaming at someone. In Croatian, direct objects always take the accusative case.