A chain and sprockets can last anywhere from 1000 to 30000 miles depending on how it is looked after. Keep it well adjusted, cleaned, oiled and rust free and it will last a long time. Neglect it and it will last no time at all.
If you don't want to spend the time on it, get a chain oiler, this will not only oil it, but if turned up a bit will kind of self clean it by flinging off oil and dirt. You will still need to adjust it from time to time, clean it occasionally and fill the oiler resevoir, but it is more of a fit and forget solution than the alternative.