The Borns Key drops from Lucious the Depraved who is located in the Cemetery of the Forsaken in Act I. There is a catch: he is a rare spawn, which adds an extra layer of randomness to his already random drop rate.
While random loot is bound to luck, and some might get it in the first or few attempts, it might take several attempts. In my personal experience, he spawned once every 5-7 maps. I killed him about 15 times.
Sometimes you won't see any elites. Sometimes a skeleton elite shows up. Sometimes Lucious the Depraved spawns. There were occasions where he spawned two times in a row. Important: Even if you see the skeleton elite, you should still explore all of the cemetery. About twice, both Lucious and the skeleton elite spawned in the same map.
One thing to note is that Lucious the Depraved patrols all the cemetery, including upstairs by the crypts to the sides of the cemetery. Search thoroughly before creating a new map.
The way I do it is: I create a map. Teleport to the Cemetery of the Forsaken. Ignore the skeletons. Just search clockwise. Teleport back to New Tristram. Leave the game. Create a fresh game. Rinse and repeat until you loot the Borns Key.
Lucious the Depraved spawns pools of poison beneath his feet, and likes to flee away during combat which means a lot of skeleton aggro when you try to chase him. Not much of a problem if you are geared well.
**Hocoga reports it also drops from Dataminer in the Cemetery of the Forsaken. There is a Adventure Mode bounty for him.
The Borns Key is a legendary crafting material to create the Born's Command Set. Thanks d3ros_wiki