It's advisable to at least check your PSU voltages before putting it to actual use, as sometimes its failure can result in overvoltage and kill chips inside a C64. Some recommend dropping them altogether and using newer replacements instead. The 'black brick' type supplies are considered especially risky. That being said, many of the original PSUs do work just fine to this day.
Most commonly you will get a black screen if the PSU fails - either the computer isn't getting the voltage it needs to boot, or getting too much voltage which damaged something inside the machine. Pay attention to the LED light - if it's not lighting up or is dimmer than usual, it's a pretty obvious sign there isn't enough voltage coming in. You can try re-seating the plug and wiggling the wire as well - often they can be the culprit instead of the supply itself. Another common power problem is a wonky power switch that doesn't make proper contact. If you suspect the switch, unplug the computer and rapidly wiggle the switch on/off multiple times. If this makes a difference for the next boot, the switch is the most likely suspect. If your C64 did get overvoltage, the smaller chips are usually the first to die, especially the RAM.
A faulty PSU does not necessarily always cause a black screen. Sometimes a marginally failed PSU can give off various weird symptoms, as you can see here. In this case the symptoms will often change depending on how long the machine has been switched on.