How can I use iCloud with 2 different apple ID's?
You can use separate apple IDs for iCloud and iTunes.
 Homesharing, iTunes Match, along with iTunes/App Stores can all be associated with a shared iTunes account Apple ID.

SETTINGS>> iTunes and App stores>> 

iCloud iMessage and FaceTime can be signed into with your own unique Apple IDs. A total of 2 Apple IDs are all that are needed between you and your dad. 

Settings>> iCloud>> delete account off one device and keep info when prompted to keep or delete.

If personal info such as contacts, calendars, bookmarks etc, have been merged with Info from other device it would be wise to clean up these items before signing into the different Apple ID.