Q: my phone broke but i somehow need to back it up without going in the phone, so how do I?

Now that is is broken, it is too late to back it up.

The time to backup is before something goes wrong...

If you set it to backup automatically to iCloud, it did so every time you left it connected to power and Wi-Fi.

To check if you backed it up to iTunes manually go to Preferences > Devices.

Otherwise, connect it to iTunes. If the phone was previously connected to it, you should be able to back it up from within iTunes without needing to use the iPhone screen.

In any case, if you synced your data (images, contacts, etc) to iCloud and/or your computer, you will be able to sync all that data back to your new phone.

If you previously synced with a computer then yo can connect to that computer and backup.  You can't do that if you never synced to a computer since when backing up to a new computer you have to access the phone in order to tap on the Trust this computer message that appears on the phone after you connect it to the new computer.

If the only thing damaged is the screen Apple can replace the screen for a flat fee, usually without losing any data on the phone.