Unlocking PS3 achievements is a bit tricky.
First, you have to unlock them on your PS4, which means you’ll need to buy them.
Then you have the trophy room store.
You’ll need the Astro’s Playroom trophy, and the trophy Room store trophies.
Here’s how to get them.
PS3 Trophies: Go to Astro’s and grab the PS3 trophy, which will unlock.
PS4 Trophies/Play Room Store: Go back to Astro, grab the game and the PS4 trophy, then go back to the Trophy Room Store.
Astro’s Trophy Room: Go through the trophies section of the store, and grab any PS3/PS4 trophies you want.
Astro will then unlock them.
Once you have them, go back into the trophy store, buy the PS1/PS2 trophies, and unlock them with PS3, PS4 or PS Vita.
PS Vita Trophies:-Go back to PS Vita and grab Vita trophies.
Vita trophies will unlock automatically.
PS1 Trophies.-Go to PS1 and grab PS1 trophies.
PS2 Trophies-Go to PSP/PS Vita and buy PS2 trophies.
PSP Trophies can also be purchased for PS Vita with Platinum or Platinum Plus.
PSVita Trophies are not available.
PSN Trophies -Go to PlayStation Network and grab PSP trophies.
This will also unlock PS1 or PS2 trophy trophies.
Note: If you buy a PS3 or PS4 title, you’ll also unlock the PS Vita trophies on the PS Network.
Trophy Room store: Go into the Trophy room and grab trophies from PS3 titles.
PS Vita store: The Vita store will also give you trophies, but they will be unlocked automatically.