There are two standard failure modes for chains: broken rollers, and worn pins/links. You check for broken rollers by looking, they should be obvious, and are not common with relatively modern chains that have not been abused by impacts. Pin and link wear causes the whole chain to go out of pitch, and ruins the sprockets by overloading the pickup tooth - the next one to take up the load as the chain comes onto it. The easiest way to check for pin and link wear is to remove the joining link, straighten the chain, and turn it on its side. Hold the two ends and measure how much it sags from a straight line at mid-span. If it is more than, say, an inch in two feet, it's time for a new chain. A new chain has barely-measurable sag.
