The Evil Within

In order to upgrade items in The Evil Within, you must acquire an item called Green Gel. Green Gel is scattered in various locations around each chapter, dropped of some enemies, or obtained while opening lockers in the safe room. When you have enough Green Gel to purchase an upgrade, you can go to the restraint chair in the caged room just past the nurse and her save game chart. Sit in the chair to access and view your purchasable upgrades for each category: Attributes, Weapons, Stock, and Bolts. Note that Bolts is not required for any of the trophies and should not be purchased if you only want to get the trophies out of the way. However, upgrading Bolts alongside the other upgrades can definitely help in your New Game+ run for The Quick and the Dead.

The chart I provided below shows how much Green Gel is required for each upgrade needed for this trophy. The trophy will unlock when you have purchased the final required upgrade.

As of patch 1.02: This exploit no longer works. You can play the game without the patch to use the exploit.
*Exploit Notice: There is currently an exploit for earning this trophy early. Once you earn 19,000 Green Gel, you can save your game and upgrade the Agony Crossbow until it is fully upgraded. After it is upgraded, push Start and reload the save you just created. Repeatedly upgrade the Agony Crossbow to max 5 times total and you should unlock all weapon related trophies. This could definitely save you some time grinding Green Gel for all of your upgrades. This exploit is only confirmed to work on Playstation 4, and may only work if you have no other weapons fully upgraded. Also note that the weapon does not have to be the Agony Crossbow and that the Agony Crossbow is just the cheapest weapon to fully upgrade for the purposes of this exploit.