Play the Dead Center campaign on easy. Pick up a melee weapon at the beginning then never pick up a primary weapon. Try to play with 3 human players. The AI in this game isn't very bright and you don't want to have to worry about them when you have a horde of zombies coming at you.
Things to remember:
-When you get below 20 health DO NOT swing randomly, if you get downed you may accidently fire off your pistol.
-DO NOT use the chainsaw! Once you run out of fuel you will have a pistol and may accidently fire it.
-Meleeing enemies away with LT is OK no matter what you are using so if you have something other than a melee weapon out, don't be afraid to press LT.
-Equipment are OK. Bile Jars, Molotovs and pipe bombs don't cancel out the achievement and they can help you out in a pinch.
-Keep a health kit and an adrenaline shot on you at ALL TIMES.
-Tanks are your worst enemy!!! Be sure to keep away from them and let your teammates kill them.
-According to ericxbox74 the mounted gun does not affect this achievement, however I have not checked this so use it at your own risk
The achievement will unlock after you complete the campaigns finale
*Note- The new DLC campaign, The Passing, also has melee weapons at the start, plus it is only 3 chapters so pick whichever you prefer.