Ive put some steps below for you to try to get your phone back and working, hopefully.
1 - Ensure that your computer is switched on with the most recent update of itunes installed & an internet connection
2 - Hold down the home button on the phone and keep held down whilst connecting to computer ( home button is the button on the front of your phone)
3 - Keep the button held down and keep your eye on itunes until a message appears saying that an iphone has been detected in recovery mode and will need to be restored agree with this message and then you will receive a summary page of the phone with the options of update or restore buttons (when message appears release button, you will use your comp to do everything now so literally just push the phone away and dont touch any buttons etc)
4 - Select Restore option.
5 - If prompted to create a back up (see image below) select the back up option.
6 - Select the Restore option when iTunes prompts you (as long as you've
backed up your device, you should not have to worry about restoring your
7 - The Restore process can take quite a while but once completed the phone will restart showing the apple logo and itunes will display the below message - either ok or leave for it to go off itself.
8 - The iphone will be finalising its restore so leave the phone connected.
9 - Once the restore has finished you will receive an option to either restore your iphone as new or from a previous back up. ** If you choose to restore your iphone as new ALL content you have added to the phone will be deleted. If you restore from a backup the majority of your content will go back on the phone.
If these steps DONT work then it looks like it could be a hardware issue within your phone. If its still within your 1year from purchase give Apple a ring your still within your hardware warrenty! But if you ring them make sure you try restoring your phone from new before you call as it could also be software issues & when you do speak with Apple make sure you tell the what steps you have tried so you dont have to go through it all again.