Is Shakespeare's Double Negative Grammatically Wrong?

The answer is that double negatives were grammatical in English (and still are in many varieties of English, though not in standard varieties).

Standard varieties of English (in every region, as far as I know) forbid them, but this is quite a recent phenomenon - sometimes ascribed to Robert Lowth in 1762.

No, Shakespeare's double negative was not grammatically wrong. According to David & Ben Crystal, the rule that two negatives make a positive was not applied to most uses of language:

[T]he strict mathematical logic was used only in a few formal styles of expression.

A double negative was just a way of intensifying a negative. (One could say that this usage closer to the mathematical formula (-x) + (-x) = -2x.)

The idea that using a double negative always results in a positive became dominant through the works of prescriptive grammarians in the 18th century, for example Robert Lowth's book A Short Introduction to English Grammar (1762). (At least, this claim is made on the page Early Modern English (c. 1500 - c. 1800) on the website The History of English, which also mentions a few similar works.)

David & Ben Crystal also point out that Shakespeare even used triple negatives.

(See also my website for more examples of double negatives in Shakespeare's work.)

Update: curiousdannii pointed out in a comment that this is also known as negative concord. (The term is also mentioned in the Wikipedia article double negative.)