How should I add Korean Keyboard on Ubuntu 16.04?

IBus is no longer the default IM framework in Ubuntu for typing Korean; Fcitx is. Hence the fcitx-hangul package was installed when you installed the Korean language, not ibus-hangul.

Either you can switch to fcitx in Language Support, or if you want to use IBus you can install ibus-hangul manually.

Then next time you log in you should find Hangul in Text Entry.

Reference: https://help.ubuntu.com/stable/ubuntu-help/keyboard-layouts.html#complex