This method seems to have been patched and no longer works. No need to vote negative, we get it doesn't work now.
There is a really simple way to get this challenge, virtually foolproof, where you can still die.
Load any level, preferably one you have played before so you know where the end is.
Now, play through the level, it doesn't matter if you die or not. Just keep going.
When you are at the end, you want to quit and exit to the main menu, like thirty seconds before the end. You can even quit when you lose control of Max as he is walking off the screen just as the level ends.
Now, from the main menu hit continue. It will put you at the last checkpoint of the level. Simply finish this one checkpoint towards the end without dying, and your achievement will pop.
It basically counts it as a new session, even though you only have checkpoint of the level left.
If you die just quit to main menu and continue again.
You just need to repeat this on five different levels. You can check your progress on the level select menu, as if you have done a level without dying you get a skull and a ribbon on the levels picture.
Please note, the challenge on the achievement screen will not update. It will always stay at 0%.