Hardware issues can be a derived from a simple hard drive replacement to a more complex issue. Usually, Power Supply Units are though to either work/not work by checking if the computer boots up or not. Mother boards are usually replaced, when the issue could be as simple as replacing the 3V battery. So here are some areas to troubleshoot. A multimeter will surely come in handy. let's disconnect all cables and start.
1) put the multimeter in AC mode and test the power outlet for voltage. expected is a non constant value of 220-230V(EU) or 115V-120V(US).
2) retrieve and check the small CR2032 baterry on the mother board for voltage. A value from 2.90V to 3.10V is expected. Anything lower than that will start to create BIOS and POST issues.
3) to do a basic power supply unit test: if it's a AT PSU (old P8/P9 connectors), just connect the power cable and see if it turns on. if it's a ATX PSU (P1 connector) we can set direct link on it, otherwise it will only turn on when connected to a MOBO. to directly connect the ATX PSU, grab a clip and connect the green pin (usually near middle) to one of the available black pins. this test tells us right away if the PSU is going directly to the trash, or may have salvation. be careful not tu connect the wrong pins, or you may damage the PSU or cause a power failure on the premisses. I usually place the clip first, then push the ON/OFF button on the PSU, and finnally, slowly place the power cable in the PSU (without touching the PSU directly).
4) now we are going to connect the power cable to the power outlet and to the PSU unit, and the AT/ATX connector to the MOBO. Still without actually starting the machine, we should check for the 5V pin for, well, 5V! connect the 9th pin to one of the black pins. know that 15-16 pins are located by the ATX connector "click" that grabs the it to the MOBO is, and the 9th pin will be on the back and to the left.
5) if the above steps are passed flawlessly, our PSU should be working and able to boot up the MOBO I usually pick up a screewdriver and then go over the power pins until the MOBO turns on (unless I can tell where the power pins are by looking at the inscriptions or board manual). you don't really have to connect the power button from the chassis. if we connect the multimeter to a yellow and a black pin (using the red and black multimeter connectors respectivly), it should read a value of 11V to 13V. anything lower than that should mean the PSU does not turn on or has faulty voltage. we should check the various yellow wires available. I did this myself and I got 11.58V.
Feel free to test other voltages according to the schema.