the quest is called ' The Black Star' the corrupted form of Azuras Star, travel to the Shrine of Azura to activate the quest. If you don't have enough gold, you can find it at Hjerim in Windhelm or at Hob's Fall Cave (west of Winterhold). The easiest is just buying it from Falion - it doesn't cost that much. Thankfully, Falion just HAPPENS to have a black soul gem.

The Black Soul Gem can be found in a few ways. Next ask him for a cure and he'll advise you that you'll need to bring him a full BLACK SOUL GEM. its called the black star and is one of the many daedric artifacts that can be obtained in skyrim. New main objective: Bring a filled Black Soul Gem to Morthal. There is a very powerful Black soul gem that can have an unlimited amount of black souls pass through it (only one at a time) without the gem disappearing. I do not recommend selling them unless you need gold or don't use the enchantment skill.

the souls level are much like the white souls level range(though I have yet to find a petty soul for a black soul gem) each black soul gem has a weight of 1 and a base value of 300 gold when empty and 750 gold when filled. Ive missed the chance to get the artifact that acts as a permanent black soul gem so Im looking for them. Though Usually Grand the souls that can be trapped depends on the level of the characters soul you are trying to capture, Higher levels like bandits or other warriors obviously means higher level souls where as lower level characters like beggars give lower level souls.

The purpose of this soul gem is for the capturing of black souls(Human/Humanoid characters)for use at the arcane enchanter.

Where to find black soul gems in skyrim