Iphone Back-up on Family Plan
How would my child back-up his Iphone to the cloud if he is on the family plan.
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See instructions in How to back up your iPhone or iPad with iCloud - Apple Support
It works the same whether or not the child is sharing your iCloud Storage plan (which is what I'm guessing you mean by "Family plan"). If you meant your family cellular plan, that also does not affect how iCloud backup is done.
More information on sharing an iCloud storage plan: Share iCloud+ with your family - Apple Support