There are few pictures here, because i didn't think to start taking them until about two steps into the process. Anyways:
1. Remove any webbing from the inside of the helmet (if you can). Mine just unsnapped and was put back in later.
2. Remove fish or other animal from any mounting it may be on. Generally they are either glued or screwed into a base, so this shouldn't be too hard.
3. Set aside fish gently, being careful not to break the fins.
Get out your drill and screws at this point. Make sure you are in a well-ventilated area, as drilling into plastic and preserved animals may produce fumes.
The piece of wood shown in the third picture is the original mount the fish came off of.