There are different flavors to each version.
I am going to classify these faults
into 4 categories.
This is the most precise, and uses the word classify to mean "divide into classes"; it is what you would use if you were writing in a scientific or technical style.
I am going to classify these faults in
This is less precise, and uses in as one might use among. There is nothing wrong with stating the sentence in this way, but there is a sense that other categories may exist among which the faults could be distributed, but that the speaker may be making an arbitrary distinction.
I am going to classify these faults as
This one is really a poor construction, in my opinion. It equates faults with categories, or implies that a transformation may be made that turns faults into categories by the process of classification. It is inferior in logic and meaning to the other two. Your first choice is the best.