You've got the right idea. It's time to dust your computer :P

Unplug your computer from the mains and disconnect anything else attached to it. Hit the power button on the computer to drain any remaining charge and if there is a switch on the rear of the computer (the power supply) switch that to the off position. This is a safety thing, of course :P

Once its all free and drained, take it to a flat, non-carpeted surface with enough room for you to work on it. Remove the side panel (some cases differ, but it is usually the computers right panel) from the case to look at the insides. Now would be the time to ground yourself too.

Once you have a peep at the insides you can start. Places like the fan exhausts and inside of the panels can be wiped, as can the blades on the fans themselves. If you have a can of spray air/canned air then this can be used to blast dust from inside the heatsinks of the CPU (large central block on the motherboard) and the graphics card. The canned air can also be used on your power unit too. If you don't have a can, then its up to lungpower. Don't get too dizzy

You may have to remove a few components to clean them properly, but don't attempt anything that you would be uncomfortable with doing. For example, removing the fan from the CPU heatsink to get better access. Consult here or use Google if you're confused about how to safely remove a certain piece of hardware.

Once you are happy with your work, just reassemble everything in the reverse order you took it apart, cross your fingers and turn it on.

As for the buzzing sound, that could either be the hard disk working or perhaps one of the fan bearings are beginning to give way. If the sound starts getting louder or suddenly changes it will be time to replace said fan. Good luck!

GMAC