Hello and welcome to Android Forums. Hopefully I can help you with your problem.
S3 owner here. Place a charged battery in the phone. Boot the phone up and give it a good 5 or 8 minutes to boot up fully. Hook the USB to the computer and then to the phone. Hopefully the computer will recognize the phone. If it does, you'll be able to see it under My Computer and save the data. If you've given the computer a minute and it's not seeing the phone, you can pretty much go ahead and consider the phone dead. The only thing that can be saved is what's on the sd card. You'll need a micro sd card adapter to access the data via the pc.
If you have any other questions regarding your device, I invite you to join Android Forums. We have many sections for specific devices which contain a wealth of information.