Left 4 Dead 2

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.

Obviously best done on easy. If you don't have any melee weapons at the start just use LT until you come across one. Doing this with three bots is usually the easiest way, as all three will use guns, making it easier to get through each level. Be careful if you are knocked down as you will automatically switch to the pistol, even if you didn't have one. So if you see your life getting low try not to spam RT, as you may accidentally shoot when downed. Achievement will unlock as soon as you finish the last chapter's finale.

ok the easiest way to get this achievement is the DLC Campaign "the passing" straight away there are melee wepons on the floor and there are ony 3 levels and btw pressing LT doesnt stop the achievement
Hope i helped :)

Here's a sped up version of this being done on Dead Center. Just make sure to only use your melee weapons and watch out for accidently firing off your pistol, and you should be good. Also, if you are playing online, let your teammates know you are specifically going for the achievement, so they have a heads up on it.

Only Use melee weapons. Axe, knife, Paddle etc. The Chainsaw is also a melee weapon and will not last forever and after its gas is gone you will drop it and get a pistol. If you do not start out with a melee weapon on the table or however the campaign starts just melee with your fists and use the computer for help till you find a melee weapon. If you go down you will automatically be given a gun and 1 shot will blow the achievement so remember not to shot if you go down. Best done on easy.

The easiest campaign to do this on will be Crash Course on easy difficulty that will be coming out soon with the Cold Stream DLC.

Crash Course only has two maps and is very quick to do, compared to the Passing which has three and a long scavenge finale.

Pick up a melee weapon from the helicopter where you spawn then use that throughout the campaign until you get onto the armored truck at the end.

Achievement unlocked.

I would suggest doing this one on the hard rain campaign. It's not long, some parts involves small number of zombies and in the last chapter you can avoid the tanks by hiding on the roof while your team kill him off. And in the start, there's a few melee weapons. So you can pick one to your liking.