For routine cleaning and maintenance, we recommend that you rinse off the coils of your chiller with water directly after each use (save the first 5 gallons of chilling output water in a bucket to dip the chiller in immediately out of the wort), empty the chiller of any excess water, dry with a soft towel, and store your chiller in a dry area.
While some normal oxidation of the copper will occur, proper cleaning and maintenance will help protect your chiller against excessive oxidation, or verdigris (a bright bluish-green encrustation or patina formed on copper or brass by atmospheric oxidation).  If verdigris does start forming, we recommend boiling the chiller in a water & vinegar solution, scrubbing it with a stiff bristle brush, steel wool, scotch pad or similar and then drying it off until no more greenish powdery oxidation remains. For more stubborn spots, you could also rub the green areas with a solution of ammonia and salt using a thin cloth or shoelace. These stubborn spots are very uncommon, however. Copper itself is naturally antimicrobial.