You'll unlock the electric glove melee weapon once you reach character level 8. Switch to the glove at a character setup screen or workbench.
Your special melee attack builds up fairly quickly over time. Once it's ready the melee indicator in the bottom of the screen will light up. Simply press "RS" to swing, if activated you'll see a brief over-the-top melee animation that should kill one zombie, and send a few others staggering. Simply do 20 times for the achievement.