How to attack a general polyalphabetic cipher?
A general polyalphabetical cipher is just a combination of several general monoalphabetical ciphers, each applied on every $n$-th letter of the message.

So the first thing is to find out what $n$ is (i.e. the key length). For this we can use the index of coincidence just like for Vigenere.

Then we can split the message into $n$ parts (columns), and try to break each of it as an individual monoalphabetic cipher, starting with frequency analysis. (We'll have to correlate information from the individual columns to do bigram frequency analysis, too.)