The Devil You Know

To get this quest flag and be eligible for Tower of Despair raid, you have to beat 4 Devil Battlefield quests.
 * A New Invasion
 * Bastion of Power
 * Genesis Point
 * Sins of Attrition

The Weapons Shipment and Wrath of the Flame is not required.

Also, Boots of Anchoring is not required to get into the raid.

Note that in order to enter the raid, you must first cross the Devil Battlefield wilderness area (inaccessible to raid groups) to the Tower of Despair cave (where the actual raid entrance is located). There, you will find an NPC who will upgrade your Amrath portal, allowing you to return directly to the caves from Amrath.