Troubleshooting PCB of an ATX power supply?
The order and method to debug and root cause the failure depends on the individuals experience and skills. A multimeter is capable of assisting a root cause analysis process to a great extend, but depending on the complexity of the circuit you need other tools too.

I am assuming that you have the electrical schematic diagram, PCB Layout and part placement drawing for the circuit. For starter the following follow might work.


Power up the PCB and check if the initial power supply stage is functioning and supplying the necessary power to the next stage.
Provide the necessary signal inputs check if the inputs are received by the subsequent stage and processed
Check to see which of the outputs are active


At this point you will have dig deeper, it is difficult to advise with much knowledge of the circuit.

Edit: Now that we know that your circuit board is an ATX power supply the simplest debugging suggestion is to check if the 5VSB (Stand by supply) is active. With the multimeter check if you have 5V on this line. Below are to images to help you. 

 


As far as I understand if you pull down the PS_ON (Green wire) the PWR_OK line will go high. This is a indication your power supply is good. Below is power supply timing sequence from Intel specification. 



Also attach some notes about PS_ON and PWR_OK




Your may find the reference below helpful too. 

Be Safe, Be careful, and Good Luck

References: 


Dangers of Old ATX Power Supplies?
Using PWR_OK to Activate an ATX Power Supply Automatically
ATX (Advanced Technology eXtended)
How to Convert a Computer ATX Power Supply to a Lab Power Supply
Standard ATX power supply
ATX power supply specification question
ATX power supply design guide