The charger port on the Galaxy S III can sometimes break , especially after extended use. In order to identify your problem, start small and work your way up.
First, try using a new (or at least ''different'') charger cable for your phone, any android charger would do the trick.
* Sometimes the charger simply doesn't seat correctly in the port, so if a different charger works on your phone, inspect your old on for any weakness in the cable, check that the securing tabs aren't broken, check that the 4 pins aren't bent or missing.
If a new charger doesn't do the trick, test the battery. Take the back off of your phone by lifting under the indentation of the phone and then remove the battery completely.
* For more detailed instructions, check out this tutorial [guide|12507|here]. Flip the phone over and plug in the new charger to the phone and to a power source.
If your phone turns on without the battery inside of it, then you just have a bad battery, which means you just need to buy a new one and replace it.
If not, your issue is most likely with the charger port of the phone on the motherboard itself. There is also a guide to replace your motherboard [guide|12696|here], but this is a bit more advanced and requires some extra tools.