The Canterville Ghost has a very rigid set of duties in the story. While we do not know the origin of these duties, they consist of a number of activities.

First of all, the ghost is obliged to appear in various parts of Canterville Chase at particular days and times. This is clarified in Chapter Four:

It was his solemn duty to appear in the corridor once a week, and to gibber from the large oriel window on the first and third Wednesdays in every month.

Secondly, it is the ghost's duty to maintain the bloodstain in the library. This is a visual reminder of the murder he committed there in 1575 (of his wife, Lady Eleanore). When Washington Otis cleans the stain with Pinkerton's Detergent, the ghost resorts to stealing, as we learn during his conversation with Virginia in Chapter Five:

It is a very difficult thing to get real blood nowadays, and, as your brother began it all with his Paragon Detergent, I certainly saw no reason why I should not have your paints. 

Finally, it is the ghost's duty to scare people. The ghost has terrified residents of Canterville Chase for the last 300 years and he looks back on his supernatural career with great pride. When the Otises fail to feel terror, the ghost becomes depressed and forlorn because his purpose for existing is suddenly brought into question. It is this failure which brings the story to its climax as the ghost leaves Canterville Chase and goes to the Garden of Death where he can rest for eternity.