This information in this Instructable is specific to the Asus UX21E, but by reading my tips and following the general disassembly lessons you should be able to work on your own laptop.
That being said taking apart electronics has the potential to damage or destroy the electronics, so don't try to repair your own computer unless you know what you're doing.
You need a couple of supplies before you begin this process:
Screwdriver set with Torx bits
Ice Cube Tray
Step 1: Take the Back Cover Off
Do Not Use the Wrong Size or Type of Screw Driver!
Take out the eight short screws and put them into the first slot in the ice cube tray.
Now take out the two long screws and put them in the second slot in the ice cube tray.
Step 2: Remove the Battery
Remove the screws and put them into the second slot in the ice cube tray.
Step 3: Remove the Motherboard to I/O Cable
Step 4: Remove the Cooling System
First remove the five black screws, and place them in the next ice cube tray slot.
Next remove the two silver screws, and place them in the next ice cube tray slot.
The cooling system will be stuck down slightly to the CPU. take your fingernail and hook it under the leftmost black screw hole and lift slowly and gently. Be particularly careful while pulling the cooling system off of the CPU.