"I don't like it either" is the most common way a native English speaker would express this sentiment.
"I don't like it too" and "I don't like it also" are generally seen as improper because, arranged this way, there's a contradiction between the negative "don't" and the inclusive "too"/"also"; the statement seems to reject and affirm at the same time. Contrary to that, "I also don't like it" and "I too don't like it", by placing the inclusive word closer to the subject, implies that the speaker is including themselves in a group that is rejecting the object ("it"). These uses are seen as archaic, but the sentences may be used in an artistic way to call attention to the statement.
"I don't like it neither" is improper because of the double negative, but if you were making fun of a stereotypical "unsophisticated" culture or dialect such as a "redneck", you might use it.