Yes, Azura's Star(and its alternate form, the Black Star) has/have unlimited refills.
Regarding how one or the other can help you level up Enchanting, here is what could probably be considered the most pertinent information.
The information in game regarding the Black Star says it can only trap humanoid souls(playable races mostly with almost no exception) but the item is bugged so that it can actually hold any type of soul(just like a normal black soul gem). The unofficial patch "fixes" this. So if you play with mods, this may work differently for you.
With that out of the way, there are a few ways to raise your enchanting skill. Probably the most obvious is to enchant items(about 300 xp per item, about 90,000 xp needed to get from 15 to 100 skill so about 300 enchantments made). Another is by disenchanting items. This is limited as there are only so many effects you can learn. Yet another is to pay for training, and another is to read skill books.
However, another way is by refilling enchanted weapons. If you obtain the Steel Battleaxe of Fiery Souls from behind Gathrik's throne in Ironbind Barrow and disenchant it, you can enchant some weapons with soul trap. Then go out and kill anything with the Black Star in you inventory. After the trapped soul fills the Star, turn around and refill the weapon with that soul.
A grand/black level soul earn you 0.6 times the XP of enchanting an item, greater souls yield 0.4x, common souls 0.2x, lesser souls 0.1x and petty souls 0.05x XP. If you have the mage stone active(XP earned in mage skill accumulates 20% faster) and a simple rested bonus(5% faster XP gain) you boost these values to 0.75, 0.5, 0.25, 0.125 and 0.0625 respectively). That means killing about 400 bandits, forsworn, mages etcetera ( or more lesser creatures). For me, after 600-1000 hours on a character I've dispatched 4000+ people, 5000+ animals, 1000+ creatures and the same for undead(which do have souls, some of which are grand level).
This way, you can level Enchanting while out playing the game instead of buying a bunch of filled soul gems(or empty ones you fill yourself) and enchanting stuff(you can still do that which can be quite profitable with the right combo if items, gems and enchantments). You can of course combine all these methods(training, disenchanting, skill books, etc). I suggest only disenchanting effects you intend to use for yourself, followers and max profit as the more effects you disenchant, the more cluttered the interface will become when you try to enchant multiple items with the same enchanting potion.