You will unlock the tier 3 abilities once you get your spear to level 5, the first time you upgrade it will be done for a main story quest. You will need artifact pieces to upgrade it further, in order to obtain these you must find and kill members of the Cult of Kosmos. Once your spear is level 5 simply upgrade an active ability to level 3 and the achievement should be yours.
Edit: Cerebro DOW says that upgrading "Fury of the Bloodline" ability to tier 3 does not count towards the achievement.