You'll get this trophy for killing 500 enemies classified as Kobolds. You'll know they're Kobolds because the top line in the enemies name will include the phrase, example: " Kobold - Hurler". However, some Kobolds, like the ones found in the Neverdeath Graveyard won't have the Kobold title in their enemy name but still count for the trophy.
The best places to find Kobolds are during the Nest Egg quest in the Kobold-Infested Sewers and in the Neverdeath Graveyard adventure zone on the eastern side the area called Craftman's Rest. The Kobolds in Craftsman's Rest are referred to as "Cult of the Dragon - Hurlers/Minions" and you'll find them in groups of 3-5, sometimes with CotD Initiates/Mercenaries, they are small and orange-ish.
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 Kobold Slayer entries.