How to backup files if your mac screen is broken

I'm assuming you had no backups prior to this failure. That being the case:

Have you tried an external monitor? If that works, then proceed with backing up. If your Mac has a removable internal disk you might can pull it and use some sort of adapter to read it. If the external monitor doesn't work, and you might need the screen to set that up anyway, then your backup possibilities are limited and your best course will be to have Apple evaluate it.