Can an iPhone be backed up when its screen has been damaged? I have an iPhone 5s and I dropped and smashed it the other day and now it has just stopped working. The screen is fully black and I cant see anything nor touch the screen but it charges and vibrates when i get messages. I want to back it up before getting a new phone but when I try and back up it asks for me to put my  passcode in which I can't do. Is there any possible way I can actually  back up my phone?.
Yes - the screen has no bearing on whether the dock connector and internals are operational. You can charge the device on a wall adapter for 30 minutes and then plug it in to iTunes.

As long as the device is working (i.e. whatever damaged the screen didn't also damage the connector and the storage), then it will show up in iTunes and let you make a backup.