"Bite me" is an expression used in a lot of TV series. Does it mean "leave me alone" or something else? Example: Someone tells Penny that she is stupid. She answers, "bite me". Is it an old expression?
It's a very informal usage of the idiom in AmE.
bite me (very informal) - used to say to someone that they have made you feel angry or embarrassed.
I don't think it's an old expression. So, if you find someone telling bite me, they are annoyed and about to burst!
If you look at this video (uploaded a year back), it talks about the slang bite me (The video just talks about the term in Hindi).
Another (vulgar) definition is here on the Urban Dictionary.