iPhone 4 only turns on when plugged in to wall, not computer

iPhone 4 won't hold a charge. It says restore when plugged into the computer. new battery has been installed with no luck. All I really need are the photos from the phone, but it is not connected to any service so I can't send the photos out. Also, the phone turns off within 2-5 minutes of use even while plugged in. Anyway to remove memory from iPhone?

i Have some memories on this phone id really like to keep. Thanks for any help.