You didn't say you were using an external keyboard. I could have made this SO SIMPLE had I known that.
When you use an external keyboard, from the external keyboard, you cannot use emoji's from the keyboard. If your keyboard has function keys on the top row, see if one near the #4 key has a picture of a keyboard. If it does, simply taping that key when in messages, brings up the onscreen keyboard, so you can add an emoji. When you're done and want to go back to the Bluetooth keyboard, tap that key again and you'll be back to your external keyboard.
If your keyboard doesn't offer the option to bring up the onscreen keyboard, you might want to check out Bluetooth keyboards which do offer this feature. I use a Bluetooth keyboard with my iPad, but always make sure it has the onscreen keyboard key.