Enchanting is one of those skills that is annoying to level and costs a lot of money to level especially if you are looking to just do training until level 90. I will outline some easy steps that you can follow which will help you make the leveling of enchanting less daunting and even somewhat profitable.
The first thing you will want to do is go to the mages college in Winterhold. There you will find a number of people that will sell you soul gems. If you are a little short on funds just buy their lesser soul gems and leave the greater and grand ones alone. personally I usually go for the common, lesser and petty soul gems because I don’t think you level enchanting any faster if you use a stronger soul gem. I advice buying both empty and full soul gems.
Next you will need a weapon that is enchanted to trap souls. Usually the enhancement will say “If target dies within X seconds fills a soul gem.” This is the type of weapon you will want. Then travel to High Hrothgar. Find Arngeir and ask him for the location for a word of power. This requires you to have done some of the main quest line. Alternatively you can go around the country side and kill all the animals you can find and go through dungeons and kill everyone there. Normal soul gems will only trap non-human souls. They will trap the soul of skeletons and draugr.To trap a human’s soul you need a black soul gem. A human is always the strongest soul possible. You can also do a quest that rewards you with a soul gem that never breaks, google “black star” to find out more. Normal soul gems disappear after a single use.
After you have filled up a soul gem find an enchanting table. I believe they are called arcane enchanter, but I am not sure. The one I usually use is in Dragonsreach in Whiterun. Once you are at a table disenchant anything you don’t want anymore, although you only need one enchantment to level up enchanting, disenchanting also levels up the enchanting skill. Usually I like to enchant armor or necklaces and rings because for weapons you have to confirm the enchantment which makes the leveling take a little longer. The best way to acquire necklaces and rings is by pickpocketing the people around you, usually they are wearing some sort of jewelry.
The best trainer I found for enchanting is Hamal who can be found in the inner sanctum of the temple of dibella in Markrath. She only offers training after you complete the quest she gives you when you rudely interrupt her in her ceremony. She will train enchanting up to level 90, but for a dear price.