how do i back up my phone

You will need to be able to approve the computer to back your iPhone through your iPhone's screen. Furthermore, the iPhone will need to be unlocked by entering your passcode. If you are not able to perform the above tasks on the broken screen then back up process will not work.

Axel F.

You'll need to replace the screen. Another idea is to bring it to a repair shop and ask if they can connect any screen for a while so you could make a backup.