I own (at the time of writing) six Commodore 64s. Three are breadbins, two are C64c longboards and 1 is a C64c shortboard, thankfully working as it's schematically quite different from the others. Four of the other five were dead or dying.
A couple of these had bung 6510s (cpu) which is easy enough to diagnose. Before having some spare 6510s arrive from ebay I took a punt and unsoldered one from a working C64 to use it as a tester. This also had the bonus effect of making the donor board a useful for testing other 6510s by virtue of putting a socket in where the 6510 sits.
One of my bung 64's had its 8701 clock fried. This was found by simply poking around with the CRO. I think the symptom was no video signal (not just blank screen). I found that my old analogue CRO and my lack of skills limiting: but just about any mug can use a CRO to see if a clock signal exists. 8701 easily swapped with another and problem solved.
The 2nd last C64 to fix was the one that was most satisfying: "out of memory in 0" came up on boot. Goggle search and lemon64 forums suggested RAM fail and directed me to World of Jani. World of Jani (http://blog.worldofjani.com/?p=158) suggested looking for flashing or flickering characters on the screen: and I had the occasional " (double quote) appearing here and there. Take the ascii of " which is 34, subtract the ascii for space (which is what is meant to fill the screen, not ") which is 32 and you get 2. That is bit 1 which, according to Jani's table, is U9 in an old 8-RAM board (since those RAM chips are only responsible for 1 bit in the entire address-space). I pulled out the old RAM, socketed it, and put in the only spare 4164 I had, lifted from my other 8-RAM-chip C64 which had one of its RAM conveniently socketed (ie replaced way back when). Bob was my uncle: C64 fixed. Thank you World of Jani.
The final C64 I fixed was a blank screen with video (called a white screen?). VIC2 was good. I suspected memory fail after testing every other chip liable to give a blank screen, so I bought a Dead Test cart rev 781220. Plugging it in gave 3 flashes in a sequence: meaning RAM U11 was toast. I bought some 4164 off ebay and 4 weeks later the dead RAM chip was replaced and this 64 was resurrected.
I have enough parts to have five working C64s, tho only two SIDs work fully (a couple half work). I'm down a 8701, a 6510 and three SIDs.
See also: resurrecting a piece of junk vic 20