Weapons are "ranked” from 10-kyu to 1-kyu and then from 1-dan to 10-dan. The "rank" is determined by experience which is gained when you destroy enemies and bosses with a specific weapon. If you use more than one weapon to destroy a boss the experience is divided among the weapons regardless of how much damage you did with each one. All weapon experience is furthermore reset to zero if you use a continue. You can see what weapon you are ranking up on the top left (1P) or top right (2P) of the screen.
Note that you have to get all 9 weapons (except the Normal/Handgun) to 10-dan, each in a separate playthrough (best done on Normal Difficulty). Be sure to either finish the game or let the continue timer run out for the game to save your progress and you’ll unlock the achievement right after you 10-dan the final weapon. If you have trouble finding a specific weapon (such as the Bomb) or simply want to get dual weapons of any kind early on, you can suicide on the first few enemies and get to collect three random weapons after you use a continue. If you didn't get your desired weapon simply repeat the process until you are successful.
More information about Weapon Master can be found HERE.