Do not fret! Curing vampirism is much easier than most tutorials online let on!
It seems that you are having trouble with enchanting a weapon with soul trap. Since that is the case, it is likely that you do not possess the enchantment for soul trap yet--so enchanting in order to fill that black soul gem of yours is not an option. Your only other option at this point is to find and/or purchase the Soul Trap Spell Tome. You can find this spell book at most all Court Wizards in the main cities, or at any of the Master Mages in the College of Winterhold.
The easiest way to get this for me has been to go to Farengar, the court wizard in Whiterun--as he is the earliest available court wizard you meet. Find him in Dragon's Reach, in Whiterun during the day. He wears blue robes and spends most of his time in the south/western room on the first floor of Dragon's Reach. Speak to him, ask what he has for sale, find his books, and look for the Soul Trap spell. Shouldn't cost anything you can't afford after selling off a few items you've plundered in dungeons. From there, go into your inventory, and read the tome. It'll disappear from your inventory, but you'll get a notification saying that you have "Learned the Soul Trap Spell". Next, go to your character menu, then the Magicka section, scroll until you find the Conjuration spell list. This is where you will be able to find and equip the spell-trap spell. Depending on the amount of magicka you have, and the perks you have in the Conjuration spell tree, you may or may not be able to cast it just yet--so be sure to take note of whether or not you can (it's not a super magicka dependent spell, nor is it a high level to use.)
Now to fill that soul gem. Once you have the spell equipped in one hand, and you are sure you can cast it, be sure to have your weapon or offensive spell of choice in the other. The way this spell works is that you cast it on any creature or person, and if you kill that creature or person within a short amount of time, it will fill a soul gem. Now because you are filling a Black Soul Gem, you need to take the soul of another human. Your best bet is to just go venture out into the open world of Skyrim and find yourself a Bandit Camp. Cast the spell on a low-level bandit, and kill him as fast as you can! You'll know that it worked if purple whisps and whatnot get absorbed into your character somewhat like how a dragon soul looks like. You can double check this by going into your inventory, and down to the miscellaneous section to find that Black Soul Gem. It'll specify whether or not it's filled. Once filled, go ahead and return back to Falion, and finish up following that quest!
I hope this helps a little bit, if not let me know and I'll try my best to guide you better! Good Luck my friend!
P.S If you would like some help on how to enchant and disenchant items, in general, I can go ahead and fill you in a little bit on that as well if you'd like. Feel free to ask me!