To get the most thorough cleaning of the whole filter, put it in an ultrasonic bath with water and some isopropyl alcohol. Leave it in the bath for a few seconds, remove, then gently wipe dry with clean microfiber cloth. The only problem with cleaning it this way is that all the oil from the filter thread will be gone too, so you want to be careful not to over tighten it on the lens.
DO NOT do this with polarizing filter because the liquid gets between the 2 layers of glass and won't come out unless you take the filter apart.
The same small ultrasonic bath is typically used to clean spectacles and jeweleries.