Use hot water to wash away the remaining bits of food.
If there are food stuck on the bottom, use a brush or a hard sponge to scrape them off.
(Optional) If it is too greasy, use a couple of drops of dish soap and wash a bit more.
Dry it out with some paper towels, or on low heat on the stove.
That's it.
A cast iron skillet is one of the toughest objects in your kitchen as is, you don't have to baby it, just make sure it doesn't rust, and the seasoning will build up naturally as you cook.
I put mine upside down in the oven to dry. We always did that when i was a kid because the pilot light would provide a tiny bit of heat to dry. Everything is auto ignite now, but i still do it and think of my mom and grandma every time.