The CIAs (Complex Interface Adapters) are among the most common chips to fail in a C64. They run fairly cool and don't usually fail spontaneously, but are directly connected to the I/O ports and so very vulnerable to static electricity - the joystick ports in the case of the CIA1 and the Serial port in case of the CIA2. To prevent damage, never plug anything in or out while the machine is powered on. Also make sure the connectors and cables are in good shape as they can cause problems too.
The CIA chips at U1 and U2 are identical, so if they are socketed, a useful test is to swap them with each other and see if the symptoms change. If they do, one of the chips is likely bad. U1 faults generally cause keyboard/joystick issues while a failing U2 tends to cause disk drive issues. Another test especially useful for black screen C64s is to just remove these chips altogether - the computer can produce some form of a startup screen even without the CIAs.
Another task that the CIAs handle are the TOD (Time of Day) clocks. Both U1 and U2 have their own clocks, which can be useful for diagnostic purposes. You can use software to test if one of the clocks is giving incorrect readings and possibly figure out if a CIA is bad.
One notable feature of a scrambled startup screen caused by a bad CIA chip is that it tends to have normal color and borders. It's not uncommon for a faulty CIA to produce a blue garbage screen, but if you are getting multi-color garbage when booting into BASIC, it's likely something else.