As Platypus suggested, if you have some soldering skills you can replace the capacitors. These are fairly inexpensive, so you won't be investing very much money in the attempt to salvage the computer.
One thing that you should always do before handling any of the components inside the case is to discharge any electrostatic charges you may have. This can be done by touching the metal of the case before touching anything inside. It should go without saying that this computer should be unplugged from the wall before doing anything.
You will need to remove the motherboard in order to replace the capacitors. If this is new to you, you may wish to make notes of the different power and data connections that are removed from the motherboard.
There are a couple of things that you will want to observe while changing these capacitors out. These are known as electrolytic capacitors. These are polar sensitive, this means that the positive lead of the capacitor will have to go to the positive location of the motherboard. There should be a + on the board indicating that it is positive.
If you are replacing more than on capacitor you may want to mark the board and the capacitor with a number to keep track of it's proper location, you can also mark the board with a + at this point for future reference.
To remove the capacitors a solder sucker will help remove the solder from the leads. Wherever you wind up purchasing the capacitors should have an inexpensive model and be able to demonstrate its use. Another item which can help is a small amount of liquid flux made for the type of solder you are using. When solder is reheated it tends not to liquefy as it does when first melted. A very small amount will help remove the old solder with the sucker. Remove the solder as best as you can from the holes. If the holes are clear you should be able to pull the capacitor out. If it's still held by solder apply the soldering iron to one lead and gently pull the capacitor away from that hole, repeat the process to the other side as needed.
When you go to install the capacitors make sure to observe the proper polarity. Hold the soldering iron against the pad which the lead is protruding and the lead itself. Wait till the solder will melt and adhere to both the pad and the lead. It takes a very small amount to make a good solder joint. I would advise not going back and trying to clean up a solder joints too many times as the solder will begin to lose its ability to liquefy.
There are web sites which you can find by Googling, I found two that cover most of what you will need to know. This
article addresses soldering technique, how to solder, what a good solder joint looks like, and what a poor or "cold" joint looks like. This
addresses installing and removing board components as well as soldering techniques.
Hope this helps.