This is easy to level up, but it will require you to procure a lot of items. Some of these items can be combined with the levelling of other abilities. For example, you can use a weak soul gem on an iron dagger. Since you will be making a lot of iron daggers to level up your smithing, you shouldn’t have to spend any money sourcing these. To make things better, the dagger is worth more after being enchanted, but don’t expect it to be a lottery win. You can purchase weak soul gems very cheaply in most stores, but you could fill them yourself using the soul trap spell. Keep spamming this and you should be able to increase your enchanting skill very quickly.
There are potions that you can drink to enhance your enchanting skill that let you perform actions outside of your base ability in order to level up even faster. If you find an enchanted weapon on your travels, you can use it at an enchantment table to disenchant. You will obtain the enchantment that it contains and it will further your enchantment skill.
Recharging enchanted weapons that have lost their charge is also a quick way to grind this without having to rely on enchanting daggers or other weapons. Keep refilling the enchantment and you will be furthering your enchantment abilities.