This is a solution for the last level - AZTEC (which is needed for the speed run). Doing this method should take you about 5 mins to complete the level.
I highly recommend iTz Canada Ehh's videos or the text game walkthrough for the rest of game.
This method presumes you know How to climb as explained in the walkthrough.
When you land in AZTEC, first move one of the large sticks under the standing up wall. Place it slightly underneath where the red bar is and pull up on the stick to make the wall fall down.
Take the stick out and take it with you to the gap in the next wall. You now need to prise this door open by sticking it in the hole and pushing the stick to the right.
Once through start climbing the steps and instead of turning left at the top. Climb straight up the wall in front of you using the cheese method outlined in the walkthrough.
Once at the top follow the grass around until you see the AZTEC temple. Climb the temple and drop in the hole.
PS. You can put your Xbox into standby mode, I did this achievement across 3 night's of play just don't play anything else, never quite to the menu and hope your Xbox doesn't update!