How to back up your phone after it is broken and the screen no longer works

There is no way to back it up now. You need a functional screen to input commands to back it up. As is always the case, backups need to be made, and kept up to date before something drastic occurs.

Contact Apple about a display replacement -> iPhone Screen Repair & Replacement - Official Apple Support

Often, if the damage is only to the display, Apple can replace that without affecting data at all. But note there is no guarantee that your data will not be wiped when undergoing any repair or service (Legal - Repair Terms and Conditions - Apple). Again, as always, backups are the users responsibility.

Hi, you can check this - How to back up your iPhone, iPad, and iPod touch

If you have backed up your phone recently then you can restore it once the screen has been replaced.

You can book an appointment on getsupport.apple.com or use the Apple Support app.

You can also contact Apple Support - Contact Apple for support and service

Sorry, but you can't. Hopefully, you have been making regular backups all along which you can restore from when you either get this phone repaired or replaced .