CPU fan and cooling  system troubleshooting
First you have to take off the computer case cover and seek for the CPU inside, that will be connected in some socket

In this particular case we are dealing with an earlier Pentium III CPU that fits in a longitudinal slot, more similar to a regular expansion slot than to the modern CPU bi-dimensional sockets (more like a square filled with little holes for the CPU pins connectors to fit in).
This makes this case particularly interesting, since we have to deal with not everyday assembles.
As you can see the power supply is obstructing the way to extract the CPU & cooling system set. Even though we could with a short precision screwdriver just extract the fan to check it alone. But we want to check the whole set.
So we will proceed to separate the power supply from the case, just by unscrewing it. We don't need to unplug it from the mother, since we'll need the power to check the system later, and it doesn't bother the process.

 
Now the way is clear to try to pull out the CPU from its slot.

Here you can see the CPU from upwards.
This kind of arrange is particularly hard to remove. It has two brackets at both sides that fixes the set in position, in addition to the pressure of the slot. To release this tops is advisable to slip a screwdriver between the CPU edge and the bracket, to push it aside and free this side of the CPU.

Keeping the screwdriver pressing the bracket aside, pull out that edge of the CPU form the slot. Repeat the operation for the other edge.

Finally the CPU complex is free.

Now is time to separate the CPU from the cooling system: fan and heat sink. For that you have to handle the metal blade on the side opposite to the fan.

Finally the CPU and the cooling system splits.

 
Here you have the 3 pieces of the complex: CPU (green board with chips), cooling system (with the fan on the top), and the metal blade that keep the together.

Now is time to remove the fan from the heat sink unscrewing it carefully.

 
Now you have the fan free to work with it. Note that we don't unplugged the power wire from the mother, since we'll need the power to check it, and that it doesn't bother at all.

Now proceed to remove the dust and dirt accumulated between the fan blades and the heat sink blade. Note: the pictures below show this process with the CPU still in place in the motherboard. Of course is much more easier and effective when you have all the pieces apart.

In the back of the fan there's a little sticker covering the axis head. Remove it gently and just enough to make the axis visible and reach it with a lubricant oil dispenser. If you doesn't have anything better use kitchen vegetable oil... (uh, how terrible!)

Now reassemble the whole set following the same steps backwards. Reinstall the CPU complex in the CPU checking it has fitted OK in the slot (otherwise your computer may not boot up, if so check the CPU is completely inserted in the slot).

Turn on your computer, and recall the BIOS menu immediately. If your BIOS has an "Advanced Features" item, which provide CPU temperature and fan speed measurements, select and monitor them. Temperature should not exceed 38 Celsius or 100 Fahrenheit, and fan speed should be between 5000 and 6000 RPM. These values may vary with the hardware model.

Monitor these values for 3 hours at least to be sure that they are stable. If so, your overheating problems are solved.