This achievement can be won easily.
Start the chapter 6-3, and play it normally, no problem if you die, just keep going.
After you kill the beast, progress into the next room.
You'll be in a small tunnel with an earth pillar at the end. While you're being chased by lava, go up the pillar.
Now pause the game and go back to the main menu.
Press continue and you'll be back in this tunnel, but the death counter is now at zero.
(The game starts a new session and thinks you are not dead so far.)
Now just climb the final tunnel, exit the chapter and you'll get your achievement.
If you die, repeat it by returning to the main menu and continuing till you finish that final part without dying.
" WARNING"
I'm not sure if it works if you have already finished the story!
Edit: It has been reported that you can't get the achievement if you have already finished the story. The game give you the chance to replay missions, but you can't continue after a checkpoint, you have to reload the entire mission.
Thank's to Ericles Oliver for that clarification.