This is why my AV receiver still works. :-)
So you start by just doing a visual scan of all the caps on a board to see if any are bulging? If nothing is apparent, where do you go from there?
The part in question: https://i.imgur.com/hZDA08y.jpg
Repaired: https://i.imgur.com/IMbBUeN.jpg
Start with a visual inspection, looking for any components which appear to be visibly damaged. For capacitors this would be something bulging, but if something has failed thermally the chip might be burned. If you see anything burned, though, I would exercise caution before proceeding.
After that, measure the output of each power supply with a multimeter and confirm that it is at the appropriate voltage level. If you find something which is off, you need to confirm that all of the additional components supporting the power supply are present and functioning. You can measure resistors on the PCB with a multimeter.
Some AV Receivers will also fail after a number of heating / cooling cycles. In this case, the solder joints on the relays have gone bad (physically disconnected) and they can just be soldered again to correct the problem.
Honestly, AV Receivers are a great place to start since they are generally both well documented and designed to be serviced.
edit: one other thing -- you can check the clock output of different crystals / oscillators, etc, if you have an oscilloscope. This can be useful to know if the clock is operating at all, and, if it is, confirm that it is clocking at the appropriate rate.
There were a few that were visiting bulging or had leaked, but I replaced all of them just in case.