This achievement must be completed on NORMAL or higher difficulty
Also, the 10 kills must come in one life if you are attempting this in campaign.
This is quite simple to do in Firefight variants such as Sniper Fight, Sniper Attack, or Crash Site which all start you with the pistol as your secondary weapon with unlimited ammunition and loads of grunts to kill in the first wave.
As an added bonus, Crash Site will provide you with extra shields and no lives so that you won't get mixed up counting kills across lives. This mode can be found in Firefight matchmaking under the Score Attack play list.