This guide is out of date, check more information from here: http://www.speedrun.com/Super_Skeleman/thread/amcul

You need to get the backflip item first, then hit a checkpoint some area where there's wall above (it can't be higher then two blocks), find a wall where there's enemies or spikes to kill you while doing walljumping, then do multi-walljump input (left side of the wall for example: hold right, smash left & jump button) 1 or 2 frames before you die at enemy or spikes and keep doing that input until you are going through the wall above the checkpoint.

If done succesfully, what game does is that it spawns you in dead state at checkpoint and same time Skeleman jumps up about 6 pixels still in dead state, then Skeleman revives from dead state (gets up) but 2 pixels of his head gets stuck in wall above him and goes up through the wall because player got stuck inside the object. You can skip pretty much almost everything in the game doing this.

Video of wall clip glitch from spikes plus inputs! https://vid.me/Q1GC

Here's GIF with slow motion that shows what happens when you do the glitch: