I did exactly what Sherwood's answer said with my iPhone 5 that died on me yesterday.
The trick totally worked, after the Apple Store told me that there is NOTHING that can be done and that I would lose almost 1 year worth of data! If you want to check if the screen is still touch sensitive, either call your number and see if you can answer the call by hitting the area that would have the "answer button", or see if you can power the phone down by hitting the top button and do a "swish" where you would normally see the "slide to power down button" at the top of the screen.
Once you start it back up make sure you do another swish on the bottom of the screen where the "slide to unlock" button would be (I forgot about that one initially.)
I found that I was not able to unlock the phone while I had it connected to my PC/iTunes. So unlock the phone first and then connect to ITunes to get a full backup. Hope this helps!
Note that you can also use the accessibility options to help you. I activated the VoiceOver options (http://help.apple.com/iphone/7/#/iph2062e901c) while my phone was plugged into my computer through itunes to help me figure out what I was touching on my black screen, and select the appropriate buttons. Do this somewhere you won't be embarrassed to have your phone read to you though.