The humming noise most often is a fan. To be sure, you have to open your case and listen for what piece of hardware is making the humming. If it is a fan, read below, if it is a hard drive or the power supply, scroll down to the appropriate section.
Typically, the bearings go bad on the fans after a long time. This can be fixed. If it is the rear or front (or side) case mounted fan, all that needs to be done is to remove the troublesome hardware and install a new fan. All you need is a screwdriver.
If it is coming from the processor, northbridge, or graphics card fan, that might be troublesome. If there is a lot of dust, try using a Dust Remover can, which sprays compressed gas, to clean the heatsinks and fans. This should be done regularly as preventative maintenance. If the noise persists, then the bearings are bad. SOMETIMES the fan is removable, other times it is not. Typically, Intel factory heatsinks have integrated fans, whereas AMD factory heatsinks have fans that are clipped on and can be removed. IF it is removable, simply buy antoher fan of the same size and attach it. For the CPU, install the fan so it blows air upwards, for better cooling, unless you have a side fan blowing down.
If the fan is NOT removable, you will need to install an aftermarket heatsink. This often requires dissecting your computer's hardware. For this you will need Arctic Silver 5 (a small tube of special compound), Isopropyl Alcohol, Cleaning cloths (or q-tips), a screwdriver, and that manual for the motherboard and aftermarket heatsink.
If you buy a heatsink that does not require additional hardware, you should be good. Typically these look like stock heatsinks, but are made by different companies. If you bought a very large heatsink (such as the ZALMAN CNPS) or a Peltier or Liquid Cooling, may the force be with you. To install those, you need to disconnect everything, remove the motherboard, and install new mounting brackets that are provided, then put everything back together.
For the direct replacement heatsinks, it's pretty straight forward. Socket 478 heatsinks have 2 levers you pull up to release. Socket 775 Intel heatsinks have 4 tabs that you have to rotate and then pull up to release. AMD Socket A (462) and Socket 939 are much like old Socket 370 (Pentium III). Just a clip that you push down and pry away with a screwdriver on one side, which when it releases you watch fly into your head, leaving a massive penetration wound. AMD Socket AM2 uses locking levers on each side, which is easy and secure. Wiggle by rotating the heatsink some before pulling up, or the processor might pull out with it. Carefully use the Iso and cleaning cloth to remove all traces of heatsink material off the processor, and apply a very small (!) dab of Arctic Silver 5 to the top of the processor. Only the top square part, not the whole top area. Use a business card or folded paper to even the AS5 perfectly flat on the CPU. Then, take the NEW heatsink and install it, reverse of removal. Make sure you lock down any levers. Follow the directions they give you!
If it is the graphics card, there are options. Determine if you have AGP or PCI-Express, and buy a newer card as an upgrade, or get an aftermarket heatsink. it's going to be just as difficult as putting on a new CPU heatsink.
If it is the Northbridge, and the fan is not removable, it's going to be a chore. Buy an aftermarket heatsink and install it, just like the CPU. You WILL be removing the motherboard for this.
This is not good. If your hard drive is making noise when it hasn't before, you have a failure coming on the way. Your best bet is to back up your data NOW, and buy a new hard drive. Determine if you can use SATA, and if you don't have it (you don't see SATA 0 and SATA 1 on your motherboard, they are L-shaped and marked so), get IDE. Find ALL your software install disks, and WINDOWS install disks, and when the new drive comes in, disconnect old drive, install new drive, format, install Windows, install Programs, connect old drive, transfer files, done, disconnect old drive, put drill through drive, and discard.
If your Power Supply is humming, this isn't immediately bad, unless it is an electrical buzzing. Electrical buzzing means imminent explosion, whereas humming means fan bearings. Either way, I can not recommend opening a power supply to put a fan, because it is dangerous. New power supplies are inexpensive. Remove the old one, check to see that the new PSU it has all the right connectors, buy it, connect it, done. If you don't want / can't remove the PSU, get one that is 20+4 pins instead of 20 pins or 24 pins. That way if you have 20 pins you can leave the +4 disconnected without consequence.
In any case, the only real expensive job is the integrated heatsink for mobo, GPU, or doing the hard drive, all of which will take a significant amount of labor and time and money to pay someone to install. The fans and the power supply are easy fixes that the mechanically inadept can do. Just remember to read the manual.