iCloud storage of photos with originals on external drive

Welcome to the Apple Community.

I assume you mean synced to the cloud rather than backed up. Backing up is a different process and best not done to the cloud.

All you need to do is move your photos library to an external drive and make sure iCloud photos is enabled.

https://support.apple.com/en-us/HT201517

In addition to Winston Churchills answer:

• check if your PhotoLibrary on the external drive is the SystemLibrary:

Photos, Preferences, General

• TimeMachine can backup the external drive: https://support.apple.com/guide/mac-help/time-machine-options-preferences-cpmh0054/mac