Quest this is related to: Orshan Barbarian C2 Advancement Quest
The quest description (the story explanation, the quest menu explanation, and the quest description pane on the HUD) all mention that you can get kills with your basic attacks in addition to your barbarian skills.
This is incorrect, as it is actually any kill with a barbarian skill regardless of what was used while fighting the mob.
If the text is correct, then this is a bug with the quest itself, but if the quest is as-intended then the phrasing needs to change to be correct.