Wait until the pan cools a little, then scrape off excess food pieces, oil and grime with a stiff spatula. Sometimes food wipes off easier when the surface is still a little warm. Sweep the debris off the edge of the surface and catch it in a rag. Be extremely thorough.
Scrub with pumice, Brillo pad or a griddle brick. Check to see the manufacturer's instructions; every type of griddle requires different maintenance. Newer griddle surfaces may need special treatment--like with recommended griddle cleaner and water, using a softer brush so the surface won't get scratched.
Remove your griddle's trays. Wash them off in hot water. Use a medium rough-edged scrubber and wash the surfaces with mild dishwasher detergent. Wash of the soap suds and then leave it in sanitized water or bleached water for a couple minutes. Take the trays out and dry them by hand. Set the trays aside for a minute.
Wipe the trays with a soft cloth to shine the pieces up a little. Dab the cloth with some club soda or a little bleach. Stroke the subtly dampened cloth all over the surface of the trays to shine them up.
Do another wipe down of the main surface of the griddle. Attach the trays back to the steel griddle. Prepare it for another day of cooking. Do small maintenance like wiping it off with a spatula between meals.