Ick.... Black-screen C64 is the hardest to test/repair.
Personally, I'd tend to suspect VIC before SID chip problem. Is it just the video that's out, or is the whole computer not completing self-test to get to the READY prompt?
(In other words, is the computer hanging before READY, or just not DISPLAYING anything?) Can you hook up a 1541 drive to it and type LOAD"*",8,1 and see if the drive grinds? :lol: As stupid as that sounds, if the drive DOES respond to the blindly typed command, you know the computer is at a READY prompt you just can't see. Then you test the VIC and probably the capacitors in the display circuit. If the drive does not respond, you know the computer isn't completing self-test, and you start testing everything that can cause self-test to fail (lots, unfortunately).