The more you use a weapon, the more experience you get. The more experience points you get, the higher level it gets. With that in mind, here are the steps to getting a weapon maximized.
1.) Level a weapon to level 5. Pick a weapon you enjoy using. (Mine was the Warmonger.)
1B.) As the game will remind you, upgrade your weapon when you can. Use raritanium crystals to do this by pressing
at the Gadgetron Vendor. Keep an eye on it -- as your weapon level advances, more nodes on the upgrade grid will become available.
2.) Beat the game on a normal playthrough. Then start a challenge mode playthrough. This will allow you to unlock additional level advancements on your weapon.
2B.) You will need to unlock the second tier of your weapon by completing the card set for that particular weapon, and you'll need to buy it at the Gadgetron Vendor after upgrading it to level 5 (using bolts).
C.) Level your weapon to level 10. Upgrade on the upgrade grid as necessary.
Once you're at level 10 and the grid is filled, the trophy will pop.