To unlock this achievement you need to deal 10,000 points of damage with unique weapons - with some exceptions.
Unique weapons added by the Gun Runner's Arsenal DLC - like Bozar or Medicine Stick - count, as do unique weapons from the base game - such as That Gun or Benny's pistol, Maria. However, unique weapons added by the other DLCs - such as the A Light Shining In Darkness or Salt-Upon-Wound's Power Fist - do *not* count.
This will likely be the first Gun Runner's Arsenal achievement you'll unlock, as unique weapons are good and you should use them. However, if you find yourself using a DLC unique quite a bit (like I did with the Holorifle in a recent playthrough) know that you're not working towards the 10k when you do so.