when backing up your iPhone, what do you mean by "make sure your screen is locked"? Press the button on the right after initiating the backup?

That instruction only applies to automatic backups while an iPhone is charging. For instructions for an immediate backup, see the instructions in the top part of How to back up your iPhone or iPad with iCloud - Apple Support

You don't lock the screen when using the Backup Now option.

For backing up automatically when charging, either use the right side button or just let the device lock itself based on the time chosen in Settings > Display & Brightness > Auto-Lock.