I'm a trained PC/Laptop/Netbook/Games Console repair person (one of my other trades) you can check the Power unit with an ordinary multimeter set to 20v DC on the Dial, pull out the IDC (insulation Displacement Connector) power lead that goes from the power supply box to the motherboard, plug in the PC and test it using the metal case as ground, you should get 5 volts on one of the ends,8 volts on another and 12 volts on another, the motherboard does not use all of them it will probably use only the 12 volt one, dont worry about the Power supply fan not coming on, this is load dependant and will not come on until the computer heats up a little in normal functioning use, after you have checked that the connector can be pushed back into the socket on the board, dont worry it only fits in one way round so you dont need to worry about reversing it...I think what your problem will be will not be the power supply unit, I think it will be the FAN on the processor, it is most likely dead, after a few years, depending on the original build quality of the computer, the bearings in the fan motor will go and it will stop functioning, all of these are now 3 wire types and the fan outputs a small signal/ripple from the third wire, this is picked up by the motherboard/mainboard and detected as a working/functioning fan..
When the fan dies and you push power on, the board begins to start up, and the first thing the control circuit does before loading the BIOS (Basic input ouput system) and then searching for the boot strap(set up instructions) on the Hard Disk is to pole (check) the processor temperature management BIST (Built in self test) circuit and the fan controller IC (this controls and monitors the fan speed depending on the load, hence faster when playing games than browsing the web) and if this returns either "no fan present" or "fan fault" then the board cuts off in order to protect the main processor (CPU) from damage, if this did not happen then the computer would come on as normal but with no fan, and the processor chip would pop in a puff of black smoke.
All you will need to do is change the fan, now these normally come in a kit with a new processor and PC shops will either try to sell you a whole new processor kit at 100quid or try to get you to pay them to fix it at 200 odd quid plus parts, silly money considering the cost of a whole new computer, so dont go near them.
take the processor fan out, write down whether its an intel chip or an AMD chip and what speed of processor it is, then you can go on the internet and buy a new fan for a tenner or a top of the range one for 20 pounds.
the Fan will come off easily as it will have two little silver lever clips on the side that tug out then lift up, or it will have 4 outsized plastic screws, also easy to take off, when you lift it up do so very slowly as there may be some sticky thermal transfer paste(like gooey glue) underneath that is there to improve heat transfer to the heat-sink and fan, it will come away but dont yank it out fast...or you will pull the processor to bits...
The small wire on the fan just clicks out of its little white connector and will go in just as easy when you fit the new fan, DONT touch the last little lever next to the processor as this is to hold it in place, sometimes there is a small screw there instead, again you should not touch this...
The fan will be a standard model and you would just go online or into a decent small computer shop and ask for it by processor type, so you would just walk in and say "Athlon Fan please" or "Celeron Fan Please" or whatever your processor model is....
you can find this by either looking at the sticker on the back/side of your PC or by going into (control panel-performance and maintenance-system) and it will be displayed there, or you can go to the start bar then (search) and type in command, click on it when it comes up, it will make a black box appear on the screen.....type in the word SET and hit enter, the processor type will be displayed on there as well, to close this type the word exit and hit enter.....
You should now have the processor model name and can get your new fan, fitting it should take about 3 minutes including getting the side of the case off and the computer should now be fully working again...