I am an electronics engineer, so trust me when I say;
No hair driers please!
Hair driers can actually melt solder, depending on solder's composition. Improper heating of solder will result in porous solder with increased electrical resistance, also known as "cold" solder. You need a microscope to actually be able to see this.
Best possible solution is to disassemble the computer as suggested on other answers. Use DIELECTRIC spray, it is specially designed to clean electronic components without damaging the PCB (Printed Circuit Board) or electronic components, you can use a tooth brush if you are extremely careful. Do not do this in a hurry, this requires a lot of patience.
When disassembling, it is extremely important to use antistatic precautions, specially if you live in a place where air is dry. Or house heating is used, especially electric heating.
At the very least you must discharge yourself by touching an electrical ground. Like your computer's chassy (if its metallic) while STILL connected with a properly grounded power cable (you must touch where there is NO paint, usually the back of a desktop would be unpainted).
If you want to be really safe you would need to buy an anti static wrist strap so that you are connected to ground ALL the time you are working with your computer.
You can get a ground connection from your wall sockets IF your sockets are PROPERLY grounded.
If you are thinking that this is all an exaggeration consider that you could have anti static shoes, anti static pants, ant static shirt and anti static gloves, also the floor you step on could be anti static.
If data on your computer is important, you can remove the Hard Drive and read its information by connecting it to another computer, even if yours does not work any more.
You can do it!
Good luck!