Not sure how to 'use' the Black Star. I have brought it to Nelacar and then fought Malyn Varen and won. Now the star is given to me. What do I do to use it?

Just like a normal soul gem except it doesn't vanish when used, you can refil it again.

It's essentially a refillable Black Soul Gem. Kill a humanoid that you've cast Soul Trap on, and it should go into the Black Star. Use the soul as you would any soul gem, and instead of disappearing, it will stay in your inventory, but the soul will be used anyway.

You can fill the star with souls that you can use to make enchanted items or recharge enchanted weapons.

To get a soul into the star you have to cast soul trap on an enemy or use a weapon that is enchanted with soul trap.
If the enemy dies while under the effects of soul trap, its soul goes in the gem.
It can hold one soul at a time.

A soul from any of the playable races will always be the largest size, a grand soul.
Bandits and necromancers etc. are good for this.

Ahh...thank you all. I guess I have to get a soul trap ability.

You have two choices. There's a spell version (you can buy it from most mages), or weapon enchantments (usually found either as random loot or randomly on merchants. You can find a "Steel Battleaxe of Fiery Souls" in Ironbind Barrow).

Actually its has one more choice. I use bound weapon (bow), and i have the perk Soul Stealer: Bound weapons cast Soul Trap on targets, and i make contract for DB with the bow, so its the easiest way i think. You can actually kill anyone with bow while sneak, and that way you have filled BS, and no bounty.