Tomb Raider

As you progress through the game, you will unlock weapon mods. These mods are unlocked through finding pieces of the weapon throughout the island. As far as I can tell, when and where the pieces show up are completely random. I advise you to search as many bodies and salvage containers as possible to find these pieces. Once you collect enough pieces, a new modification will become available. You will be notified when you find pieces and complete a new modification via screen prompt. Visit a campfire to complete the mod, and then use salvage to upgrade the weapon.

There seems to be a few exceptions to having to find pieces to get an upgrade. For example, you will be given a new bow automatically through playing the story.

Once your weapon is at it highest possible modification state, all you need to do is purchase the remaining upgrades. After buying the last upgrade for that weapon, this achievement will unlock.

The final form of each weapon, and how many upgrades are needed to complete it are listed below:

Competition Bow - 10 Upgrades
Combat Shotgun - 9 Upgrades
Commando Rifle - 11 Upgrades
Magnum Pistol - 10 Upgrades

If you have any further questions or ideas on how to make my solution better, please let me know via PM or comment below. Thanks!