After spending around four days trying to do the Speedlunky achievement, I finally did it - and it feels good. Probably one of the hardest achievements I've unlocked.
For those trying to do this achievement, I would recommend not trying to rush. I lost so many lives from rushing and not taking an extra second to survey my surroundings. Also, try not to keep you finger glued to the run button.
Don't worry too much about item pickups - although always break crates that are nearby. The number one item to obtain is a compass. It will make your run much quicker and is essential for the Ice/Temple areas. One or two extra packs of bombs and ropes will probably be enough. A cape is also very useful, but not essential - although makes the Ice area a breeze. On a few occasions, I managed to obtain a jetpack, but I personally don't think it helped much.Try to save damsels if you can, but don't go too out of the way.
Follow the set path for the first two worlds - Mines/Jungle. Work left to right/right to left and look for the place to drop down. If you see a sacrificial altar or Tomb trap treasure, the correct path is usually the opposite way. These areas usually result in a dead end. If a damsel is near a sacrificial altar, sacrifice them - you may get a cape, compass or climbing gloves - all very useful for a speedrun. In the Mine area, always keep a jar/stone in your hands to set off traps and avoid wasting a rope. For dark levels, always keep your torch with you. Even if you have a shotgun, take the torch. Drop the torch down gaps and follow the lighted areas. Also, turn the in-game brightness up all the way.
For the Ice area, always look down before jumping/dropping. Try to keep central if you don't have a compass. As you look down, keep an eye out for the iced doorway. You may not always be able to see the door from where you are, but you may be able to see the iced arch. With a compass, work your way down carefully. With a cape, float down slowly - although still be careful!
Bombs will make your life considerably easier in the Temple area. If you have a compass, bomb your way to the bottom. The most dangerous enemies here are the Tomb Lords (mummies) and the Crusher blocks. I think I died more from Crushers than anything else when I got to the Temple. Try to avoid them, but never go near them if you are in an enclosed space. If you have spare bombs, blow them up if they are in your way. Also, try to avoid a jump over a lava pit.
Olmec. I was squashed 3-4 times by Olmec before finally getting the Speedlunky achievement. There is definitely an element of luck involved in the final stage - particulary with the monster spawns and where they fall. However, here's what I did. As the stage starts, move forward. When Olmec squashes the birdman, run up to his left side. He will attempt a squash. As he rises, run under him and touch his right side. As he rises, move under him to the space on the left. Keep moving left and right as he attempts to crush you. As you get towards the bottom, send up ropes either side of Olmec - this way, you will be able to jump to the rope you are nearest to when his final slam takes him into the lava.
As a guide, aim to do each level in the first two areas in about 30 secs. Try to complete the areas in the following times:
Area 1 - Mines - 1.45-1.55
Area 2 - Temple - 3.45-3.55
Area 3 - Ice - 5.10 - 5.20
Area 4 - Temple 1-3 - 6.30 - 6.45
You need at least 45 seconds to defeat Olmec.
Don't restart if you're a little over these times. You never know what the next level will be like. You may get lucky and have a straight drop to the exit, saving several seconds.
For a visual run through - see xX Jolly's excellent youtube guide above