To find all 42 artifacts is a simple task, complete enough missions that you can earn
Once that's done, you'll have all the remaining artifacts on the map, there's 42 in total, collect them all.
There's one more step however, you must "listen to the memory" of every item. All you need to do this is to click on them (the ones with the exclimation point next to them should still need to be done. Ones with a check mark are already "listened to". Then rotate the object with the stick til you feel a little rumble, and see a blue spot. once you find the right spot the text will change to say press Square to listen to the memory.
After you hit Square, you can rotate the artifact all you want, but the dialogue will automatically let you hear the memory with out needing to find it again.
The best way to do it is to find the memories as you find each artifact, it automatically switches you to the artifact mode when you find one, so you might as well discover the memory as you find the artifact.
You do NOT need to listen to the memory in full, just going into the memory's screen and quickly backing out of it is enough, as long as you have a check mark next to every artifact (and every artifact allows you to go directly to the memory with square when you look at them) the trophy should pop automatically.