Iphone 6 black screen of death?
If it's just your screen that is not working properly, as it appears from your description, you can use a screen replicator like Reflector 2.  

You can connect your iPhone to your computer and use the screen representation to perform an iCloud backup.  


  Reflector is a wireless mirroring and streaming receiver that works great with Google Cast™, AirPlay™ and AirParrot 2®. Mirror your content to the big screen without wires or complicated setups. Play games, watch movies, demo applications or present from the palm of your hand. Even send your screens directly to YouTube for others to watch live. 


If your device needs to be replaced, you can then restore your data from the iCloud backup.