To get this achievement, you need to complete the majority of the level without dying. I believe the measured period for dying begins when you pull your first lever dropping a bridge.
This level can be very tricky, as the controls can be a bit sensitive, the camera angles can shift suddenly, and some jumps (particularly with the stone smashers) can fail even if you aim perfectly.
A few suggestions:
Different jump-grapple-jump combos require you to press the first 'A' either a short or long time to properly complete the combo.
When the stone smashers are around, be sure not to hurry your jump. Sometimes when the stones are reopening, a jump perfectly placed will still fail. I found myself jumping onto the perch point and bouncing off into the pit, but when I delayed slightly to allow the stones to fully open, I was properly aimed onto the perch and landed without issue.
Always press 'Y' when jumping to a ledge, perch, etc. A number of jumps on this level will trigger lost balance/grip with high regularity. Just spam the 'Y' button.
In the room with three mummy creatures, scale down and shoot them from the lowest ledge. They can be a bit tricky if you just jump down.
Lastly, and this should be obvious: Take your time and make the jumps properly. Line up your camera angles, face Lara the right way, etc. This isn't a time trial -- it's all about accuracy, and you'll need to be careful on this level.