I will call somebody else, not neither him nor you

I have a sentence

I will call neither you nor him

I guess it means

I won't call neither you nor him

Out of it it's not clear whether I will call anyone at all but I want to say that I will call but among the people I will call there will be neither you nor him. Can I say it like

I will call somebody else, NOT neither you nor him

or should it be like

I will call somebody else, neither you nor him

I am worried NOT neither you nor him = 2 negations which is a bad grammar.

I will call somebody else, neither you nor him

is better; as you said, it avoids a double negative. Though it's worth noting, this isn't actually against the rules of grammar, it's just a style rule. (See here)

Personally, I'd put the "somebody else" after the negation to give it more emphasis:

I will call neither you nor him--somebody else.

I will call neither you nor him. I'll call somebody else.

I will call neither you nor him, but somebody else.

It's totally up to you though. Use whatever sounds better.