What is the fastest way to get to enchanting 100?

If you have Dragonborn, follow this process:

0. Make sure you have at least a few thousand septims.
1. Go to Halted Stream Camp, just north of Whiterun. Kill everyone, grab the guaranteed Transmute spell tome, and mine all of the iron ore.
2. Transmute everything into gold and/or silver (levels Alteration).
3. Make a lot of jewelry (levels Smithing).
4. Enchant the jewelry with the most valuable enchantments you can (levels Enchanting).
5. Go to Tel Mithryn on Solstheim.
6. Alternate between selling enchanted jewelry to Neloth (levels Speech, and nets a lot of gold) and purchasing training from him (levels Enchanting, expensive at high levels).
7. Buy empty or filled soul gems from Neloth, as you prefer (levels Speech). If you buy empties, you will need to fill them yourself (levels Conjuration).
8. Go back to Skyrim, mine up more iron ore from all of the locations where it can be found and start over at step #2.

You want to level up the other skills in addition to Enchanting so that your character level increases more rapidly, since you can only purchase five levels of Enchanting per character level. If your character is dirt poor, then step #6 is much more tedious and may be impossible, since Neloth only has 1000 septims and training costs are often higher than that.

Eventually, you will hit 90 Enchanting, at which point Neloth will refuse to train you any higher. You will need to earn the last 10 levels yourself.

Some additional advice:

• Grand souls make the best enchanted gear. You may want to save a few for gear that you actually plan on using.
• Lesser and Greater souls provide the greatest amount of recharging per unit weight (assuming that all soul gems are filled to their maximum capacity), so if you use an enchanted weapon, you might want to set these aside for that purpose.
• Petty souls can be used to make Waterbreathing or Muffle equipment of the same quality as you would get from Grand souls, because those enchantments do not scale. At higher enchanting levels, however, you're better off using Petty souls for Fortify Sneak or Fortify Archery items, as those are more profitable. Always check the displayed value of each possible enchantment to select the most profitable option.
• The sidequest Unearthed, from Dragonborn, gives as its reward a set of armor with the unique property of improving your enchanting skill by 10 points, if four pieces are worn. This is a temporary effect that goes away when the armor is removed, and can be used to boost your enchanting all the way to 110.
• There is also a well-known exploit in which potions of Fortify Enchanting can be used to enchant better Fortify Alchemy gear, which can in turn be used to create better potions of Fortify Enchanting; there is no cap and so you can make ridiculously powerful and/or valuable items this way. As Skyrim is a single-player game, it's up to you to decide whether or not this exploit is appropriate to use.