The problem was that I could import contacts from my Mac to iPhone 6 but I couldn't do the opposite. I have backed up to iCloud and tried to do it that way but it still hasn't worked.
Describe which method you attempted and at which step did it fail. However, here's a working method that I used when I got my Mac.
The contacts option must be enabled and all the ones from iCloud will be automatically imported into your Mac. If this is the method you used but didn't work, please let me know.