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The size of the soul used to enchant an item determines the base charge, the magnitude of the enchantment strength and your Destruction skill determines the charge drain. In addition each enchantment has a base cost that affects the charge drain and Absorb Health has one of the higher base costs (almost 3 times that of a Fire damage enchantment).
And just to add something else in here, if you wear Destruction cost reduction gear, it will make you use less charges on your weapon. If you are wearing 100% Destruction gear your weapon won't spend any charges, remaining "infinite" as long as your gear is on.
No. Collecting souls is easy enough to do by enchanting a melee weapon and bow with Soul Trap or Fiery Soul Trap, especially if you have The Black Star. You should also have plenty of coin to simply buy filled soul gems from court mages and general goods traders.