For this achievement you need to play 6 mutations, the mutations can be found in the Left 4 Dead 2 menu under their own category. Mutations are currently being updated and changed once every 2 weeks and so at this rate it will take a minimum of 6 weeks for you to unlock the achievement.
However that being said there's a couple of major issues for players working toward this achievement. The first of which is that Valve introduced what are known as Community Mutations do not count toward your mutations count and so in short, if a Community Mutation is put up while you working on your achievement, it won't increase your count and you'll have to wait for the next mutation and hope that it's an "official" mutation. The second piece of bad news is that duplicate mutations do not count twice, for example if you play "The Four Swordsmen (Of The Apocalypse)"
and later down the line play it again, it won't count. So in short each mutation can only increase your total count one time.
This former problem has been verified by Valve in the L4D blog. (Thanks https://www.trueachievements.com/Cartoon+Ghost.htm
After our recent poll about community Mutations we received many emails pointing out that you cannot earn Mutant Overlord playing custom mutations. Don’t stop with the Mutation ideas, we will come back to community Mutations. We are just taking a little break for now.
The latter problem is also highlighted in the same blog.
The Passing introduces a series of weekly game mode events dubbed "Mutations." These are week-long runs of new gameplay modes that range from "Realism Versus" to "Chainsaw Massacre". Each week one new Mutation is featured and the previous Mutation retired. All you have to do is play 6 DIFFERENT Mutations, not 6 straight, from the cycle from 6 different weeks.
For those interested the blog can be found here
Now, as stated earlier the game updates the mutation once every 2 weeks. Every Friday the in-game blog will notify you of the mutation change (approximately 15:00 Eastern/Noon Pacific).
There have been reports of people getting stuck at 5/6. It appears the issue is that Realism Versus
only counts as .5 (0.5/6). To solve this simply join a game of Realism Versus
again and play for a while. During the game the achievement should unlock. This "fix" comes from Ben LegendTo receive credit for playing a mutation you only need to enter the challenge/game.
This has been confirmed by multiple people and is useful if you don't have a lot of time in a given week to play.
A list of mutations can be found here.
This is the most complete list available.