No second chances
To obtain this achievement you must make it to the end of chapter 4-2 named Molten River without dying.
You can try this at Entrance A or unlock Entrance B. If you want a challenge you can try A or you can attempt the easiest way to get this achievement by unlocking B. To unlock Entrance B you need to find the alternate exit in chapter 4-4 to get the Alternate entrance in 4-4 you need to find the Alternate exit in 4-5.
The alternate exit to 4-5 is located towards the middle of the map, you will see a giant hole in the ground with rocks to jump across it, instead of going across it, you go deep into the hole until you come across a a row of rocks and lava that keeps going up and down(deeper into the hole), there will be a saving point at this location.
You must go straight down when the lava drops, and stick to the left side, there will be a opening which you want to go into. Hit the switch inside the opening and prepare to go back up to the saving point, make sure to save it to make sure not to die. Now from the saving point you want to head straight to the right going across the rock platforms and the door will be open since you hit that switch below. Go through a few rooms with jumping and you will end up to the Alternate exit giving you the new entrance for chapter 4-4.
Loading up the Entrance in 4-4 you simply continue until you find the alternate Exit, its not hard to find just follow the arrows, once you get to it you unlock the alternate entrance to 4-2!
Once you load up Entrance B for 4-2 for the achievement its quite short, it took me 3 attempts once I got this entrance. I recommend using the Attractor gadget, used in the video is upgraded 1 time, not fully upgraded for those who want to know. Also the video doesn't start at the beginning at Entrance B, it does cover the hardest parts (towards the end of the chapter).
Good luck and have fun!