Back up my boyfriends iCloud to my phone because he lost his phone

You could but it would be incredibly complicated and you would then have a mixture of his and your stuff on your phone with no way to keep them separate. It is best to wait until he has a new phone and can restore his backup to that.

This support article has instructions on how to restore an iPhone or iPad from an iCloud backup -->

Restore your iPhone, iPad, or iPod touch from a backup - Apple Support

Crazyeyes126 wrote:

How to back up a different I cloud account to my phone

You don't.

If he has done regular backups, he can restore his new phone from his backup.