The LCD could have been damaged during the process of puttign the case in. Chances are slim but possible. Can you try connecting the phone to the computer and see if itunes detects it. http://www.dashrepairs.com
Can you try to connect the iPHone to the computer and see if itunes detects it. Try puttting it in DFU mode. Did the iphone stop working completely after the battery dies, could symbol a faulty battery. http://www.dashrepairs.com