I have a chain checker tool, but I just regard that as a rough approximation.

Standard wisdom: You should measure your chain across 12 outer links, from center of rivet to center of rivet. If elongation goes past 1/16", then replace the chain. If you let it get to 1/8", you've ruined the cassette and maybe the chainring too, and all should be replaced at the same time. You'll need to periodically change the cassette, too, because as it accumulates wear, it will cause new chains to wear faster after each successive change. The more you keep on top of chain replacements, the longer the cassette will wear. Extra note: measure a couple of spots along the chain. One spot may wear more quickly than another.