With the difficulty as "time trial," you have to complete the levels: Mountain Caves, City of Vilacabamba, The Lost Valley, and Tomb of Qualopec to unlock this achievement.
While playing, you can save time by:
-Using the golden shotgun cheat that you unlock after completing the Lost Island time trials. The shotgun will kill all enemies with only one shot.
-In Mountain Caves, when you get to the first tunnel that has the dart traps, don't worry about turning off the switches on the wall. You won't lose much health by just running through.
-In The Lost Valley, after getting the first cog, go up the ledges towards the waterfall. You will be on the left side of the waterfall. Now just follow the stream to find the second cog.
Also, after you place all of the cogs in the wooden thing, simply jump into the pool at the bottom of the waterfall, don't worry about climbing down each ledge.
Below are great videos by gunj17 to help you complete the Peruvian time trials.