Although the VIC-IIs are not particularly prone to failure, they are nearly always socketed and thus easy to rule out as long as you have spares. An erroneous startup screen is common, but can also cause a black screen. It's also one of the few chips that can cause a total loss of video signal when bad. In some cases the basic graphics are fine but sprites are incorrectly displayed. If everything else works but sprites in games and such are messed up, the VIC is a likely suspect. Due to its high running temperature, the symptoms of a faulty VIC often change as the chip heats up, for example garbage that gradually fills the screen from one side to another. Graphical glitches that only arise after the machine has been on for a while may be caused by an overheating VIC chip. Another factor that points to a failing VIC are graphical errors that extend to the border areas, but as these pictures prove, sometimes a failing PLA can also mess up the borders. Generally if you get an incorrect startup with flashy/colorful/animating video output, the PLA and VIC-II are prime suspects - especially if the borders are also affected.
If the power LED lights up but you get no video signal at all, check the fuse. On the older "breadbin" type C64s, a blown fuse prevents the VIC-II from working. This results in a totally dead machine with no response to blindly typed commands either. If the VIC stays cool while the other chips heat up, that's more proof that it's the fuse - the VIC is not heating up because it's not getting the voltage it needs to operate. On the newer C64C models, the VIC operates at a lower voltage and is not affected by a blown fuse. A fuseless C64C will boot up but audio will be missing and datassette won't work.
The VIC-II is usually the hottest chip inside a C64, so it may be a good precaution to add some extra cooling. They can heat up to about 70 degrees celsius, which is the specified max temperature for most chips inside a C64. Adding heat sinks and/or a fan can help if you are worried about the temperature. You can also remove the cardboard shield that covers the mainboard to help with the air circulation.
Keep in mind that it's normal for these chips to run very hot. So even if you experience problems with a C64 and the VIC-II runs hot enough to hurt your fingertip, you shouldn't automatically assume that it's the culprit. Unless it gets that hot within seconds of startup - that's abnormal for any chip.
Most C64 games have some sort of graphical glitches and quirks in them, such as flickering graphics or sprites. Make sure it's not normal behavior for the game in question before suspecting a VIC fault. If you try to play PAL games on a NTSC C64, this can also cause all sorts of graphical glitches.
CAUTION: The newer 85xx revisions of VIC chips are NOT interchangeable with the older ones. They use different voltages and you may damage a 85xx VIC if you swap it in the place of a 65xx. However, not all hope is lost even if you do make this mistake - some people have reported that the chip did in fact survive just fine.