Get an empty Great Soul Gem, and a scroll of soul-trap. Go to one of the red clothed shrines outside of Dark Fissure (east from the shrine of Vaermina), Fort Istirus (east of Kvatch), Fort Linchal (north east of Kvatch), or Wendelbek (at the headwaters of the Panther River). Go at night. Wait 24 hours at a time until the day that you see a faint blue light from the moon. Wait until the light is on the shrine, then quickly put the grand soul gem into the shrine (It's a box) and cast soul trap on the shrine. Tada: Black soul gem. There are a few other ways, but those are plot spoilers. Another easy way is at higher levels (14ish?) you can get them from Necromancer bosses and chests near necromancers.

But don't go to the mage guild with a black soul gem. It's proof of necromancy. Black soul gems can capture NPCs, like humans in towns or on roads and whatnot, making them quicker to fill. They're useless once you get the Azura's star, though.