Keyboard does not appear when adding a new contact
Keyboards on iPhone 6 are know to be buggy. My suggestions would be:


Update to the latest iOS version which is 8.4. Do this in Settings &gt; General &gt; Software Update.
If you are using third-party keyboards, disable them in Settings &gt; General &gt; Keyboard and use the default keyboards.
Try closing the Contacts app and then restarting it. (Do this by double pressing the Home button, swiping up the Contacts app, pressing the Home button and then opening the Contacts app again.)
Restart your iPhone. (Hold down the power button on the side for a few seconds, swipe to turn off, wait for your iPhone to completely turn off (can take up to a minute) and then holding down the power button for a second or two until the Apple logo appears on the screen.)