Yes, but it requires some specific pieces of hardware, an adapter and some patience from you.
- A Wifi connection with access to the internet
- Enough space on iCloud to accept a backup
- An HDTV
- Lightning to HDMI adapter.
- HDMI cable connected to HDTV
- Set the TV to accept input from the HDMI cable you'll be using for this operation.
- Connect the adapter to the iPhone and the HDMI cable to the adapter.
- You should begin to see a feed from the iPhone on the HDTV.
- At this point you can use your manual dexterity to
- Use the passcode to unlock the iPhone
- Access settings>iCloud>Storage & Backup>Back Up Now
It's tricky, but it can be done.
to get your screen working again try holding the sleep button (the on top of the iphone) and home button at the same time for 15 second you screen should go black the you will see the apple logo dont worry nothing will hapen to your iphone in the procces after it done reboting you should be on the lockscreen