Revenants are created by a burning need for revenge, stopping at nothing to get the vengeance on the one who did terrible things to them.  The revenant monks have a similar burning desire.  Their drive towards vengeance causes them to pick up abilities similar to those of their namesakes.  Once they reap their vengeance, the Revenant Monk will not die, but once their blood cools will start looking for the next person or creature who triggers their vengeance.

 

(This also could be the seed for an entire adventure, make sure to talk with your DM / GM when selecting the story for your first target of vengeance.)

Hunted Target

Starting at 3rd level, once per long rest you can declare a hunted target.  The target need not be in sight, but they must have slighted you at one point in your past and you have sworn vengeance against them.  Against this target you gain a damage bonus to your unarmed attacks of 4d6 bludgeoning damage.

Revenant Drive

Starting at 3rd level, you have the drive of a revenant, the unending drive to stop at nothing, to stop for nothing until they get to their quarry.  Luckily, the monastic training of mastering the mind and body, keeps them from dying.  You stop needing to eat, drink, breathe or sleep.

Vengeful Tracker

Starting At 6th level, when you are tracking someone who has wronged you, you gain proficiency in survival (tracking) checks.  You also are not slowed when tracking that target.

Durable Hide

Starting At 11th level, you add your proficiency bonus to your armor class when unarmored.  This counts as a natural armor modification.

Regeneration

Starting At 17th level, you gain the ability to regenerate.  You regenerate 10 points per round.  If you are hit with either a fire or radiant attack, your regeneration does not trigger on the next turn..