The best way to tell if Kimchi has gone bad is to smell it and look at it.
When Kimchi has gone bad the Kimchi will smell more sour than usual and also likely have mold on the Kimchi.
If you notice any mold on the Kimchi you should throw it out because it's bad and could make you sick.
It's normal however for Kimchi to have a slight pungent and sour smell to it but if it smells really sour then it's bad and should be thrown out.
Kimchi will last for around 4 days in the refrigerator and if it's been longer than 4 days since the Kimchi has been in the refrigerator then throw it out.
If you want to keep Kimchi longer than 4 days it should be frozen instead.
When frozen the Kimchi will last for up to 6 months before it goes bad.