You'll need to manually switch to Analog in the Options>Controls menu. The way this works is you need to hold down the Left Trigger, then use the Right Thumbstick to attack. If you want to chop off a head for example, pull the thumbstick all the way to the left/right and then quickly flick it in the exact opposite direction. It takes some getting used to. This is best done early in the game when zombies are slower and have less health.

A small trick to getting this with ease: kills while driving or shooting with analog controls turned on count toward this for some reason. You can avoid using this control mode at all if you get all 150 by running zombies over or using guns. There may be a glitch where those types of kills stop counting; to reset this issue, simply get a melee kill with the controls on and they'll start counting again