When connected to the account you want to GET data from, Go to Settings>iCloud and turn all data that is syncing with iCloud (contacts, calendars, etc.) to Off.
When prompted choose to keep the data on the iPhone.
After everything is turned off, scroll to the bottom and tap Delete Account. Next, set up a new iCloud account using a different Apple ID and turn iCloud data syncing for contacts, etc. back to On. When prompted, choose Merge. This will upload the data to this new account.
Note that this only affects the "Apple data" like contacts, calendars, reminders, etc. Many third party apps also use iCloud to store data files there. These files may be lost in the process, unless the apps also keep the data locally on the device.
NOTE: Photos in the photo stream (if you use it) will not transfer to the new account. It is advised that you save the photos to a computer before performing the account switch.