There's not much in a VIC-20, so there's not much that can go seriously wrong. 
Check the usual easy stuff first -- do a careful visual inspection of the logic board. The 6502 and the VIC chip are typically socketed, and Commodore used very cheap sockets. Simply (and carefully!) removing and reseating any socketed chips may fix the computer. It's been so long since I opened one up that I don't remember if the RAM (all mighty 4kB of it) is also socketed, but if so, you know what to do. Just work slowly and carefully so that you don't bend any leads while reinserting the chips.
Check for capacitor leakage, as you would any old electronic thing.
Also verify that the RF modulator (if that's how you're hooking it up to the TV) has clean connections to the VIC. Double-check the connections to the antenna, and make sure that the TV is tuned to the same channel that the modulator is.
A thorough visual inspection will quite frequently pinpoint the problem. The semiconductors themselves rarely rot.
Btw, here is a link to some nicely-scanned schematics: http://www.mainbyte.com/vic20/manuals.html
Check the usual easy stuff first -- do a careful visual inspection of the logic board. The 6502 and the VIC chip are typically socketed, and Commodore used very cheap sockets. Simply (and carefully!) removing and reseating any socketed chips may fix the computer. It's been so long since I opened one up that I don't remember if the RAM (all mighty 4kB of it) is also socketed, but if so, you know what to do. Just work slowly and carefully so that you don't bend any leads while reinserting the chips.
Check for capacitor leakage, as you would any old electronic thing.
Also verify that the RF modulator (if that's how you're hooking it up to the TV) has clean connections to the VIC. Double-check the connections to the antenna, and make sure that the TV is tuned to the same channel that the modulator is.
A thorough visual inspection will quite frequently pinpoint the problem. The semiconductors themselves rarely rot.
Btw, here is a link to some nicely-scanned schematics: http://www.mainbyte.com/vic20/manuals.html