In between uses, be sure to clean off the griddle surface with a metal grill scraper or broiler master brush with carbon steel bristles. The scraper or broiler brush will work perfectly to clean off any remaining food particles. Be careful, however, as to not scratch the griddle surface.
For a detailed cleaning at the end of the day, turn the griddle off and let cool for a few minutes. After the griddle surface is cooled to 'warm', clean the surface with screens or pumice with the grain on the griddle surface. Take the grease trays and empty and clean them, then return them to the griddle. Some people may 'bleach' the griddle surface with club soda or vinegar to keep a new, clean look. Most importantly, however, is to re-season the griddle once you have completely cleaned it.