Your hero is one of six classes, each a three-stage line with its own ultimate. The CLASSES window opens from the Equipment screen; the class soul it gives you sits in the soul slot and levels with your class.
1Open CLASSES from Equipment
On the EQUIPMENT screen click your name and class at the top left - the CLASSES drop-down - to open the class cards.
2The cards and the aptitude radar
Each card shows the line's three stages, its ultimate and a small radar of its Attack / Defense / Speed aptitude. Your active class is lit; the locked ones carry a price in Scrap or Ink, and unlocks are account-wide and permanent.
3Upgrade the class soul
When your class level reaches the next stage, UPGRADE appears on the active card. Click it to advance the line - Mage to Sorcerer to Archon - which upgrades your ultimate and your class soul with it.
4The class soul in its slot
Back on the Equipment screen the class soul sits in the soul slot under your portrait, with your class level beside it. It cannot be dismantled or devoured; it grows as you fight.
Screenshots are taken from the live game by tools/tutorial_shots_b.py (dfmc-client) and regenerated after interface changes. The rules behind each screen: classes, system-sockets.