This milestone quest requires you to eliminate opponents using your melee attack. Basically you need to do the finishing blow with your pickaxe.

This is difficult if you attempt to run into every gun fight with just your trusty pickaxe as you are likely to lose every battle.

The strategy for getting this quest done is to play the duos, trios, or squad battle royale game modes. You need to knock a player down then finish them off with your pickaxe so it counts as a melee elimination. If you melee an enemy your teammate was responsible for knocking down then you won't get any progress towards this quest because it will count as an assist and not an elimination.

It's best to do with friends so you can give them a heads up that the knocked down player is yours and not to finish them off.