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Build Guide

General Playstyle
Unhallowed Essence is one of the easiest sets to play in the game. Multishot hits a huge area and can basically oneshot 1-2 screens. The idea is to find the first elite pack in a rift and get the In-Geom cooldown reduction rolling, after that you should have no trouble to keep Vaulting infinitely with the low Preparation cooldown. In-Geom will also make Smoke Screen free with basically zero cooldown, which greatly improves survivability. Make sure to keep up the Focus (one generator cast every 5sec) and try to pick up most gold and all health / progress globes, which should be very easy with Avarice Band.

Build (Cold vs. Fire MS)
In previous seasons Multishot - Arsenal (fire rune) was generally considered the best, though that has changed. What makes the Wind Chill (cold rune) superior is the its slow effect is applied before the damage is done (unlike Thrill of the Hunt or Iceblink for example), and this means you can fully benefit from the Cull the Weak (20%) / Bane of the Trapped (~40%) multipliers on your first attack. If you combine this with Ambush (40%) you will almost certainly bring every monster below the Dead Man's Legacy treshold (50-60% hp) and hit it twice even on the first attack, which will usually result in a oneshot kill on basically any trash monster and sometimes even on elites, something that the fire rune fails to do consistently when the rockets are not hitting the targets you want. Counting the AoE component alone, cold will do roughly 60% more damage on the first attack compared to fire, and this is not even considering the added bonus of 15% crit chance you get from Wind Chill, which makes these oneshot-crits much more consistent. Generally you can expect to only need ~half the amount of attacks to deal with most stuff in a rift, though you will spent a couple more seconds on the RG and occasional elites here and there.

Items
Full UE set with Yang's + DML, F/R and Harrington for more damage, Nemesis Bracers to cut time spent in rifts. Obviously the crit stats and cold damage are preferred, after that stack CDR (first priorty) and RCR (second priority). I suggest to run Vitality over Multishot damage on the helm because we have a lot of additive damage buffs already anyway. The preferred amulet would be a Hellfire Amulet with cold, chd, cc, socket, as a fifth passive I suggest Single Out, Sharpshooter or Hot Pursuit. The Dead Man's Legacy quiver has been reworked and now rolls with up to 100% Multishot damage, a huge buff to the set. The 4piece (from 20 to 60%) and 6piece (from 15 to 20% per discipline) bonuses got buffed, too, and Vengeance now increases our damage by 40% as well.

Variations
There are some things you can change once you get stronger or in case you dislike parts of the build. For the cube you could run Dawn in the weapon slot and only use 3 CDR items, which will ensure 100% uptime of Vengeance. In the armor slot you could switch to Pride's Fall for more consistency, Goldwarp for toughness, Warzechian Armguards if you have trouble keeping up or Leoric's Crown when you're in full endgame gear (paired with a topaz you can get quite a chunk more legendaries). In the jewelry slot you could run Convention if you're too weak, or Obsidian Ring to help with cooldowns.

There is an option to replace F/R with SoJ and Nagelring when you are strong enough to kill everything like that anyway, like that you'd never have to spend time casting a generator and could squeeze out some more magic find, too.

I definitely would not recommend to run UE in high Greater Rifts because the set is just far too weak to compare, but if you want to do it anyway, the cold variation should work fine in groups, fire would probably still be the preferred choice for solo. Just stack a little more toughness and replace the movement speed / utility stuff with more damage.