From Destinypedia, the Destiny wiki
Enter House of Winter's den in the caves.
Push through hostiles in the caves to find the Archon Priest.
Obliterate the hostiles barring the way to the Archon Priest.
Use Ghost to break the lock on the hatch.
Survive long enough for Ghost to break the lock on the hatch.
Explore the furthest reaches of the Fallen lair to locate the Archon Priest.
Eliminate the Fallen harboring the Archon Priest.
Eliminate the Taken harboring Nixis, Hunger of Oryx.
Call Aksor the Archon Priest out of its confines and prepare for a fight.
Approach the Darkness anomaly and face what awaits you.
Slay the Archon Priest.
Defeat Nixis, Hunger of Oryx.
Winter's Run is the first Strike to have the boss be replaced by a Taken variant, depending on the version of the Strike. The second Strike to do this is The Nexus