I broke my screen and now it’s black. How do I back it up to the cloud so I don’t lose everything?

If the damage is just the display, they usually can replace just the display without having to re-install iOS and wipe the device. So your data would be intact after the new screen was installed. I just wanted to be clear that Apple will not guarantee that - but a straightforward display replacement usually leaves the data on the device intact.

And I should have asked - do you have it set up for automatic iCloud backups? If so, leave it plugged in for 24hrs so it can do an auto-backup. You can check the time stamp on the latest backup file from your www.icloud.com account

You cannot back up to iCloud nor iTunes without a working display for input. As is always the case, backups need to be made before something breaks. Now, you’ll have to get the display replaced first. Often that can be done by Apple and your data will be fine. But there are guarantees with any service by Apple about data (their service terms explicitly assume no responsibility for device data) - again, it was up to you to safeguard data before seeking service.