The Umbra is unlocked by playing a cumulative total of 350 units. While this can be achieved with any clan, using the Hellhorned as your starting clan will make it easier, as they have various imps, which are designed to be fodder units that are played - and killed - quick. For reference, I unlocked this clan in 7 runs, most of which made it to Seraph.
To speed this up, you will want to check the Merchant of Steel for the Endless upgrade to put on an imp. Not only does it speed up the units for this achievement, but it's also a great strategy for imps; when an Endless unit dies, it goes to the top of your draw deck instead of being consumed. Not only can they be played every turn to repeat their summon effects, they also absorb the first hit of damage on the floor.
Note that if you do this, try to avoid Welder Helper, as they are the hardest to make work. Either you'll have to be committed to using Torch to nuke your own imp, wait for a Sweep unit, or play it up front and let the enemy chew through 21 HP instead of 1.