They can stand alone as sentences:
I do not like biology. Nor do I like chemistry.
Therefore they are independent clauses.
They can also work with a comma as you suggest, or a semicolon:
I do not like biology, nor do I like chemistry.
You could use a dependent clause to express the same thing:
I do not like biology nor like chemistry.
(Valid, but awkward in this particular case. A comma would be valid but not necessary here, and probably improve it)
Or just simple conjunction on the nouns:
I do not like biology or chemistry.