Hi Ozan,

You would not normally use ni ...ni that way.

Je ne mange pas de viande et je ne porte pas de cuir non plus 

is a more elegant and correct way to say the sentence you mention. 

You might say using ni...ni -

Je ne mange ni viande, ni poisson = I don't eat either meat or fish

or

Je ne porte ni cuir, ni daim = I don't wear either leather or suede

I hope this helps!