You need 15 Weapon Repair Kits to fully upgrade a weapon.
You also need to unlock the Gunsmith III ability (unless you want to upgrade a science based weapon in which you will need additional skills.) Because of this I recommend upgrading the Pistol, Shotgun or GLOO Cannon.
To upgrade a weapon you highlight it in your inventory and press
to open a little menu.As long as you have Upgrade Kits in your inventory you will have the option to upgrade the weapon.
If you wan't to get this achievement done earlier you can craft Weapon Upgrade Kits, you can find a Fabrication Plan in the Machine Shop in the Hardware Labs. In the back right corner on the Ground Floor will be a caged area, simply use a Recycler Charger or Leverage to move the crates blocking the entrance and the plan will be on one of the tables in the back.