icloud account on new iphone

First be sure the you have at least 50 MB of free space on your phone (required to back up to iCloud). You can check this in Settings>General>Usage.

If you do, and it still says you don't have enough storage, connect your phone to your computer and check the capacity guage in iTunes to see how large "Other" is (hover your mouse over the yellow segment). It it's very large (multiple GBs) you may need to restore you phone to reduce it down to a more reasonable level (1-1.5 GB).

Where are you looking on your phone when you say you iPhone shows that your iCloud is empty?

And did you check the size of Other in iTunes?