First step: acquisition.
The Essence of Wrath is more openly known as the Ark of the Covenant. To acquire this deadly weapon, you must become notorious within the depths of Hell by causing enough chaos and death to warrant the full wrath of the satanic forces. Once your notoriety is full, you have a chance of two things happening: the souls of all around you are sacrificed to summon an Arch-Duke, or you shall see Hell's top agent appear before you (as Genki). He will be equipped with the Ark of the Covenant, so all you have to do is kill him for it. He has the most health of anything you'll find out and about in New Hell, so you might end up needing to unleash more than simple bullets into him. Slay him and claim your prize in the form of one of the Sin weapons.
Second/final step: slaughter.
Now armed with the Ark, you might want to either invest some Wages in Unlimited ammo for Special weaponry (14,500 wages / Lv6 for +25%, 25,000 wages / Lv12 for +50%, and 42,000 wages / Lv20 for unlimited ammo). The Ark comes with its own sub-challenge to this too, namely using
to suck up souls (similar to a specific weapon you'd find in the Ratchet and Clank games). Specific souls are required for this, as you can't just use Husks to gather kills; the basic grunts that you find throughout New Hell in retaliation to you causing chaos can only be absorbed. Dark Inciters and Legionnaires, Arch-Dukes, and Genki cannot be absorbed. While you only need a small amount for this challenge, if you upgrade the Ark, you can cause more chaos through killing a target with
. Once killed, an additional burst will spread out from the target's exploding body and seek another to ravage.