Why my phone won't charge?

So last night I restored my iPhone 4, and now i cannot let it charge. The charger isn't working. I try to plug it in to a wall charger and it won't work either. i tried my PC computer & i tried my iMac and it wont work either or connect to itunes.

and I have the other problem with my phone...

my phone keeps turning off. for example i can use it for Facebook , twitter, etc. however, it keeps shutting down when i click on Snapchat... i take a picture i try to post it and its off now.. or i go to my camera i try taking a photo and it turns off..

Is it a battery?

what about my charger??

please help..