Straight forward, just get kills with a pistol. I did this on Survival mode (on easy) so they just kept coming at me and feeding me kills.
You can view your progress here:
Pause > Service Record > Stats > Kills > Total Pistol
Tip: Don't use the Welrod, change your loadout beforehand. I used the Webley MKVI, It's a one shot kill on easy and you really don't need a silencer.