The new backpack update that categorizes your weapons, how do you get to the skins to activate? They are no longer listed on the right with shields and grenades.
There has been no such 'backpack update that categorizes your weapons'; the option to have your backpack contents sorted by category has been available since the game released.
Go back into your inventory, and look at the various listings for what buttons you can press; one of them will be to change the way it's sorted (I believe this is Pg Up and/or Pg Down on PC). One of the sorting methods is to sort by gun category, which hides non-gun items you're holding. Switching to something else should make any skins you have in your inventory become visible.