This milestone quest requires you to eliminate enemy players using pistols. Although the quest name stays pistols you can also use revolvers for this as well.

You can do this naturally in battle royale game modes but you will likely be at a disadvantage in most battles because the enemy players will be using much better weapons for the scenario you are in. They will likely use a shotgun if you are close and an assault rifle if you have some distance between you.

If you do decide to do this in battle royale modes I would recommend finding a makeshift revolver and crafting it with mechanical parts to give you a mechanical revolver. If you can get an epic or legendary version of the mechanical revolver it will do big damage.

You can also wear down your opponents shield and health with other weapons and finish them off with your pistol.

A different approach will be to jump into a team rumble game. If you can get a powerful revolver with full ammunition you can drop in close to the enemy and you will likely get one hit kills because people don't tend to use health and shield items in that game mode. When the storm circle is small near the end of the game it is usually chaotic. If you have multiple pistols in your inventory you will likely never have to reload because you can switch weapons.