Items can be upgraded using various recipes, the full list of recipes depend on the type of item you're upgrading.
You can upgrade items using one of the following recipes:
Two Equal Items
Mix two equal items to get an item one level higher than the used item. For example, to upgrade Amulet of Fire to level 2 you should place two Amulets of Fire level 1 to the mixing table.
Four Equal Items
Mix four equal items to get an item two levels higher that the used item. For example, to upgrade Blessed Shield to level 3 you should place four Blessed Shields level 1 to the mixing table.
Time required to upgrade will be the same as upgrading two shields of level 2 to level 3, so you can save a decent amount of time using this recipe.
Upgrade With Charms
Mix one item with a charm of the same level or higher. Different items require different kinds of charms - some can be upgraded with basic charms (i.e. Water Charm) while others require advanced charms (i.e. Bless Charm).
For example, to upgrade Witch Amulet from level 2 to level 3 you'll need any charm of level 2 or higher, but to upgrade Necklace of Skeletons from level 2 to level 3 you'll need one of the advanced charms of level 2 or higher.
It is important to know that the level of the charm you're using to upgrade an item won't affect the resulting item, so it doesn't matter if you're using charm of level 2 or level 10 to upgrade an item to level 2 - the result will be the same. Therefore, it is recommended to use charms of the least possible level which fits.
When you add an item to the alchemy table for upgrade all charms which can be used to upgrade an item are displayed in the ingredient list, including those you don't have right now.
You can click the missing charm to open the item details dialog and find out how you can obtain an item in the game.