Maybe the phone have a power problems (battery or else.) Try to connect it via computer with USB cable. If the phone don't want to turn on you are for service. There are not so much ways to get your information out of the phone. If the screen is out (only display) you may connect it via computer and get your pictures.
Watch over your warranty and tell the service that you need your information back. Good luck buddy
p.s. Just in case... enter in your google account via Google chrome or else browser. Watch over your Photos (they should be synced into your acc then are on your phone). If nothing there service should help alot