After completing the PK quest and given the Berserker word, you can start the PKK side-quest that has you fighting other PCs outside of the arena. The quest would have introduced you to the premise of PKKing by encountering Battle Arenas out in the Areas. This side-quest requires you to PKK fifty times as well as completing the Blacklist (which you can check on in the northern alley near the dock in Mercenary district of Mac Anu).
The Blacklisters appear randomly during Battle Arenas and, chances are high, you'll defeat all of them well before completing the 50 PKKs. Each one will have a lengthy introduction followed by a flash of their name and title placed over them before the fight. It would probably be better to wait until you are a max level (50 during Volume 1) to start taking them out as they are max level as well.
A quick journey into a Field Area would be given through (Delta) Starting Your Berserker. The Battle Arena can be found directly south of the platform. If all you're wanting to do is encounter PKs, you can finish the arena, go north back to the platform, return to town, and go back to your Warp Record to hop right back into the area.