There are two methods to getting all equipment upgrade achievements.
Standard Method:Playing through the game as it was designed, ensuring you collect as many tools as possible and pieces from secret chests etc. I'd recommend getting the Instruments upgrade to level 3 early, so you don't need tools and you gain extra pieces during your playthrough. You'll need at least 6x tools and 875x pieces to get all 5 equipment achievements.
Cheese Method:Play through the game and stockpile materials (don't spend them) until you have a total of 6x Tools, 175x Pieces and unlocked the Crossbow (Chapter 5). Now find a crafting bench and fully upgrade your alchemy to level 3. This will unlock the "Pocket laboratory" achievement. Before backing out of the crafting bench press
and "Restart Section" (Alt+Tab on PC) - this will send you back to a previous checkpoint with all your materials still intact. Simply go back to the same crafting bench and upgrade the Crossbow to level 3. Rinse and repeat until you have all 5 equipment achievements.
Sling Maxed: 6x Tools 175x Pieces
Crossbow Maxed: 6x Tools 175x Pieces - Found in Chapter 5.
Alchemy Maxed: 6x Tools 175x Pieces
Gear Maxed: 6x Tools 175x Pieces
Instruments Maxed: 6x Tools 175x Pieces