When it comes to the TTL chips (the smaller ICs excluding RAM), the model makes a big difference. Normally they run cool and are quite reliable, but the MOS versions of these chips are actually known to fail quite often. In case of a fault, check if your C64 has any smaller chips with "MOS" imprinted on them, and if so consider them possible suspects. Otherwise you may want to focus your testing on the larger chips and RAM first - unless the symptoms clearly point to one of the TTLs. If any of these chips gets particularly hot, it's a likely suspect.
A garbled or otherwise incorrect startup screen seems to be a common symptom if one of these various logic chips goes bad. The machine typically does not freeze despite the erroneous video output, though a bad TTL can also cause an unresponsive black screen. Bad TTLs tend to be difficult to diagnose since they are often hard to distinguish from other faults such as bad RAM, PLA or VIC. If the startup screen text and/or cursor is drawn twice, U14 should be a prime suspect.