In the bottom right of the screen you have a circular "meter" that fills up when you catch/dynamite fish in the fishing zones. When it is full you will be notified to push Y to activate your special weapon. Do so and you will nuke the fish and the achievement will unlock.
To get the bar to fill up quicker, when you go into a fishing zone, use X to use your weapons to catch more fish quickly, and as such, fill up the bar faster.
Good luck!!