I did find a nice place to grind this.
Very close to the start of Mission 2, you'll see one of the devices which allows you to change your weapons.
Make sure you have the Centurion Pistol in one of your two weapon slots.
Shortly after this, you'll go through a set of blue doors, and you'll see a Chrono-Echo. You'll then go up the stairs. You can go to the right for some ammo, or to the left, where you see your first Zek of the game. Fire some shots at it and it will teleport away. You can then go down some stairs, where you can get some Centurion pistol ammo on one of the boxes. There's also an Audio Recording in this room.
You will now drop down into the next room, and get a checkpoint. This is important! As you move forward into this room, a Zek (a short, bulbous-headed, light-blue-coloured enemy) will appear. This is brilliant, because you've just had a checkpoint. Kill it with your pistol, and reload the checkpoint. Keep reloading the checkpoint and killing the Zek for the achievement to pop. You'll probably only have to do this 10-15 times, dependant on how many enemies you've killed with the pistol so far in the game.
Hope this helps and is a good spot for people to use the pistol, and get this weak-weaponed achievement out of the way :)
- Dwaggie