UPDATED 1st March '17: The patch is rolling out, and this is obtainable now.
To unlock a class, you need to fully complete a quest line from one of the kingdoms. For example, Karakoth will allow the player to select a Sorcerer. When unlocked, go to "Hero" button at bottom of the playing screen and press RB to navigate across to Class. You should find the available classes here.
If you already have a class equipped, simply go to "Hero" > "Class", and press A when the cursor is highlighting your currently selected class.