A quick way to clean up your grill is to get a clean dishrag or washcloth, wet and put it on the grill right after the food is done. The steam from the wet rag will help soften the hardened "bits" making it easier to clean up after your meal.
Make sure the rag is very wet, almost dripping with water. You can also use those white shop towels that Sam's Club or similar stores sell. They look like large washcloths and hold a lot of water.