My process is generally:
1. boot up the board and verify voltages. identify what it's doing wrong so you have a base line, try flexing the board slightly, wiggling the edge connector, putting pressure on SMD and socketed chips and see if any of these cause a change in behavior. if there is no sound play around with the volume knob, and see if there is at least an amplifier "buzz", touch the pins of the amp chip and see if it amplifies the interference from your finger. If there are no signs of life at all see if it will boot in to test mode by holding the test button down during boot, or if there is a reset button if it will do a warm reset.
2. inspect the board, look for any faults: scratches that could have damaged traces, cold solder joints, missing stickers over EPROM windows, dirty edge connector, battery or capacitor damage, rust or other oxidation, cold solder joints. A lot of these things may be super minor so if anything is even slightly suspect or has a hint of not being perfect... investigate it.
3a. if no overtly obvious faults are there then I just cover the basics, use compressed air to blow off any lose dust, reseat any socketed chip and inspect the pins for dirt or oxidation, and I'll clean the edge connector whether it needs it or not (I use liquid deoxit and a nylon scratch pen to clean edge connectors). I'll always do this no matter what.
3b. If you did find some suspect items then investigate those further, I'll reflow any suspect solder joints, verify continuity across any potentially damaged traces, replace any obviously damaged caps and clean up any corrosion damage and further access those areas. If eproms are missing window stickers then put a new sticker on then dump the eprom and verify it against MAME.
4. repeat step 1 and re-access if there is any improvement. If not then try to determine where the fault is: bad graphics? no video output? bad audio? no audio output? unresponsive controls? no boot at all? game runs but with bugs? Search to see if others have had similar issues and see if you can find repair logs for that game that mention similar faults, see if there are schematics available, then start using testing equipment to test the chips in that area.
Generally speaking I'd say 75% of the time I've been able to fix PCBs just in steps 1-3. I can't even tell you how may PCBs I bought non-working that the only thing wrong with it was a dirty edge connector that didn't even look dirty.