The processor in the Commodore 64 computers runs at a fairly modest temperature and is usually not considered the first suspect if problems arise. If it does fail, it can cause all sorts of symptoms, but perhaps the most common is a simple black screen. Like a semi-faulty PLA, a bad MPU can cause certain programs to work incorrectly even if the startup screen looks normal. Garbled startup screens are less common, but also possible.
The MPU is responsible for controlling the datassette, so if you are having trouble loading cassette programs but everything else seems to work fine, it could be the MPU. But before suspecting that, keep in mind the datassette has to be adjusted correctly for the programs to load properly. If you suspect the MPU and have a datassette available, you could try plugging it in and switching it to PLAY before turning on the C64. There is at least one instance where this trick managed to help a C64 with a bad MPU to boot normally.
Another tip is to listen to the high-pitch noise coming from the audio output during switching on. It normally lasts for about half a second on startup then goes silent. If the noise continues indefinitely, it means the boot sequence is never completed and could mean a problem with the MPU. Note that you may need to turn up the volume to be able to distinguish this sound.
Since the MPU is closely connected to the datassette, make sure you don't plug/unplug the device while the C64 is powered on, as this could potentially damage the chip.