The difficult portion with this section of the game is the slide-jumps over some spikes. The tip that helped me is on the second of those jumps, you need to basically be IN the spikes before you jump in order to land your roll and survive.
Rolls don't need to be timed, hold LT during the jump and you'll roll on landing.
This video shows the entire sequence that causes problems.
http://blog.naver.com/kisikai666/130174858935Video credit to kisikai666
The achievement unlocks shortly after that video ends as you reunite with Rat, but nothing else should cause any problems.