You'll get this trophy for killing 500 enemies classified as Goblins. You'll know they're Goblins because the top line in the enemies name will include the phrase, example: "Frost Goblin - Skullcleavers".
The best place to find Goblin is in the Icespire Peak, with greater concentrations in the earlier areas.
To check your progress on this trophy first press the Options button > Quests > Journal > press R1 to go to Achievements > Slayer > hit
to show them if they aren't visible and scroll down with R-stick to the Goblin Slayer entries.