To upgrade a weapon you will need to enter a Teletraan 1 store located through out the game. You will need to purchase all 18 weapons first, or just purchase one weapon at a time and upgrade them as you go. Each gun requires a different amount of upgrades to complete it. Primary guns requires 4 upgrade each and Heavy weapons require 2 upgrade each. To upgrade your weapon you will press and buy whatever upgrade you want.
NOTE: The Electro Bolter and Glass Gas Cannon do not require any upgrades & the Sling Shock and Retro Blaster (from the GameStop preorder bonus) do not count towards this achievement.