Hi,
First check the battery voltage with a multimeter. If the battery is less then 3.8V, try to charge it externally.
When you have a charged battery, try to switch-on it. If the phone switch on normally then you have a problem with the charging.
If the phone can't switch-on, try to enter the phone into Android System Recovery menu or in Download mode (Volume Up+Home+Power or Volume Down+Home+Power). If you can enter this modes, maybe the problem is into the SW. You can reset it from the Android Recovery menu or to reinstall the firmware from the download mode. The battery must be at last 80% charged!!!
If no, try to connect the phone to the PC by holding the volume down button.
If the battery is charged and if the Windows can't recognize connected device, then you have a hardware problem.
Regards.
P.S. You can buy a JIG dongle for entering Download mode for Samsung Galaxy S3 or you can find easy in the internet how to make it by himself.