My iphone won't turn on

I have the Iphone 4 and i just randomly shut off. Like when I plug it into the wall it'll take a couple of minutes but then the apple will appear and and then it will say UPDATE IOS6 and then "please plug in to i tunes" but when i unplug it from the wall to plug it into my computer it just goes black. Once it's in my computer Itunes doesn't recognize it and my phones just black. I've updated Itunes and everything. Someone please help I don't even know where to take it to get it fixed. Apple? At&T? If so do I bring my computer? Please I'm begging you!

You can't update iOS without the phone being plugged into the wall.

You should take it to apple. If you take it to AT&T they would just end up shipping it to an apple repair center so your better off taking it to an apple store, as they would likely be able to fix it there and if your lucky you may get a new iPhone for free