Enchanting cries out to be roleplayed. It could be a kind of magician's engineering, or a more refined cousin of alchemy. Although you could certainly play an enchanter as another sort of magical mad-scientist, the profession actually lends itself well to a more gentlemanly (and sane) approach, where experiments are not so much about creating some sort of autonomous monster or mind-controling love potion of serene bliss, but rather altering the nature of things to do what they never would have done previously.
Enchantments have a huge role in mythology and literature. Cinderella's fairy godmother turned a pumpkin into a stage-coach with an enchantment, Hogwarts School's "Sorting Hat" famously talks to students who wear it, and the One Ring even contains the soul of Middle Earth's lord of evil personified. All these are enchantments in which ordinary items are magically enhanced so as to reflect some aspect of character development or plot in the story, and a roleplayer at the keys of an enchanter character can work similar magic in telling his own story.
Enchanting doesn't always change things in such obvious ways, however. Most of the time, enchanters in Azeroth simply pursue their work with magic as a kind of craft, making sharp swords pulsate with magical energy, or imbuing gloves with magical strength. They learn to draw out the magical power from one item and then weave it into another, and often work together with blacksmiths, tailors, or other craftsmen, to produce items of value to all the adventurers in business throughout the world. An enchanter may have started out thinking he would eventually make vehicles out of his garden vegetables, talk to his clothing, and literally put his soul into his favorite piece of jewelry, only to find out that all he'd end up doing was selling double-plus speed boots to swashbucklers who seemed like they were having all the fun in his place.
You won't find the really exciting enchantments in the World of Warcraft tradeskill crafting lists. For that, you'll have to dig deeper into the forbidden realms of collective human imagination. You don't want to go overboard, but to a certain extent you can get away with your own personally invented enchantments to go along with whatever stat-increasements and sparkling glow effects you want to place on your gear.
Drawing on myths, but not copying them
It would be hard to argue that you changed a pumpkin into a stage coach in WoW, but if you wanted you could claim to have enchanted a small toy bicycle so that it turns into into a speedy mechano-hog when you speak the magic words. You could enchant a magical talking sword (like Kring, from Terry Pratchett's Discworld series) or whatever other item if you like, but be careful about this option -- the sword you intend to entertain your friends may end up annoying them much more than you think. You may also make or discover items which are somehow possessed by great evil, like the One Ring, but beware that this choice may end up looking like Mary-Sueism to some of your friends, unless this great evil comes across as something more than just a copy of Tolkien's story.
As an enchanter, you might possess a ring you imbued with the spirit of your dead great-grandmother, who, aside from being quite evil, was also a very good cook; whenever you wear her ring, you smell like freshly baked apple pie. You might enchant your gloves so that they point north whenever you snap your fingers, thus helping you find your way whenever you get lost and freeing you of the need to carry a compass. An egotistical enchanter might make his cloak billow in the wind -- even when there is no wind, or when the wind is blowing in the opposite direction -- just for the visual coolness of it all. A necromantic enchanter (i.e., a death knight) might enchant his rotten old rag doll so that it follows him around all the time, wishing it were a real boy, and could point out to those who mistake his little Pinocchio for an actual zombie that the head and body are not at all proportioned like an actual human, regardless of how much his reanimated toy wants to be one.
Finding inspiration in the game itself
No matter how you try and use this profession to enchant your roleplaying experience, be sure to always follow the principle that whatever cool idea or effect you have in mind, you must find some way to make your dream come true in the game itself. Often it's best to use items and spell effects already present in the game for inspiration. If last year's Midsummer Festival toy, the "Brazier of Dancing Flames" reminds you of the story of Pygmalion, you may come up with a cool story about a mage who used to make magical sculptures out of fire, until one day he made a sculpture of a woman so beautiful that he fell in love with it and inadvertently brought it to life. You may roleplay that he sits in front of his fire woman and watches her dance for him for hours on end, wishing that she could speak to him, or go on a quest with some friends to the top of a tall mountain, where her spirit may be liberated from the confines of her brazier -- or, you could even undertake some quest to give her an actual body and let your friend create an alternate character in order to play her role (although perhaps with unintended consequences -- this fiery woman might have been a lot more pleasant before she gained a real body and the ability to speak!).
As one final side-note, enchanting is a profession that works well if you want to bring some magic into your otherwise non-magical character. It is the one profession most obviously tied to magical ability, and yet it can be practiced even by classes with little to no magical ability whatsoever. Although rare, it is certainly not unheard of to see a battle-hardened warrior or rogue who specializes in enchanting. As far as game mechanics are concerned, enchanting tends to favor casters, since it allows people to craft their own wands and make wizard oils, but if you wanted to imbue someone like a rogue or warrior with magical power in a roleplaying context, taking up Enchanting would be a great way to do it.