My iPhone's storage is full even after purchasing iCloud subscription

Increasing the amount of iCloud Storage you have will not free up space on your phone. If you are running out of free space on your phone's local storage, you will need to delete things from the phone.

See here:

What's the difference between device storage and iCloud storage? - Apple Support

How many GB do you have let on your iPhone? To find out goto General>About and look down until you see

Capacity

Available (This is what is left on you iPhone to store data)