Okay, so a lot of us here know how to properly bleed/fill our cooling system, but it's a terribly painstaking thing to write the whole procedure out every time someone needs help with it. Here is the right way to fill your cooling system.
Before you go crazy trying to fix an overheating car, make sure that your cooling system is in good shape.
Is your thermostat working properly?
Is the water pump spinning freely with no lateral motion or grinding noises?
Are your hoses in good shape, not bulging or cracking at the connections?
Is your expansion tank/radiator leaking from a small crack that you can't easily see?
Is your expansion tank cap old and maybe leaking or not letting pressure escape?
These are all eventual problems that our cooling systems suffer from.
If all the components check out and you still have problems, you most likely have air in your cooling system. Plumbers refer to this as "Air-bound."
The same thing happens in home heating systems. If you have baseboard heating in your house then you might have felt the heat on the left side and right side but not in the middle. Air can block the flow of water and cause cold spots. The same thing happens when air is trapped in your engine.
Here is the procedure for filling your cooling system to eliminate the possibility of air causing a blockage, and leading to an overheated (read: destroyed) motor:
Start with the engine cold!
Remove the small black bleed screw from the elbow that connects to the expansion tank.
Remove the coolant filler cap
Turn the ignition on but do not start the engine
Turn the climate control to the highest temperature and put the fan on the lowest setting
Slowly pour coolant into the expansion tank until coolant runs out of the bleed screw hole
keep pouring coolant until you no longer see any air bubbles in the coolant coming out of the bleed screw hole
Replace the bleed screw and check the coolant level in the expansion tank
Replace the expansion tank cap and start the motor
Allow the car to idle until normal operating temperature is reached
Watch carefully to make sure the temperature gauge doesn't rise above 12:00
If the gauge starts to move past 12:00 shut the car off immediately, let it cool and start over
After a few minutes of idling at normal temp it is safe to assume there is a minimal amount of air in the system. Take the car for a quick test drive, carefully watching the temperature. If all is well congrats, you have completed your task, if not, you may still have air in the system. Shut it down, allow it to cool and try again.
Hope this helps out.