This prompt is wide open because you can write a critical analysis paper about this story from many, many different angles. On one hand, that is good. On the other hand, it can be very frustrating and overwhelming to have zero guidance on a particular assignment. If the critical analysis is entirely up to you, I recommend an analysis of themes or characters. Often, those two subjects will blend together in a natural way; you will use characters to illustrate and explain themes, or you will discuss themes that are focused around particular characters.
A possible analysis for this story could be an examination of the cultural differences that the story displays between the British upper class and the Americans. Wilde presents the Otis family as very different from Lord Canterville. While the British might be willing to accept the ghost and the dismal weather, the Otis family is not willing to accept either of those things.
Another thematic angle is the theme of forgiveness or atonement. Sir Simon is quite guilty of killing his wife and of being a bad person in life. He continues to act this way in death as well. However, he is tired. He desperately wants to gain his eternal rest, but he is prevented from being able to cross over. He needs help, and that is why Virginia Otis is such an important part of this story. She is the "golden girl" whose prayers and presence can get Sir Simon to his eternal resting place. Your analysis could even tie this theme to the previous theme. Virginia Otis is the only member of the Otis family that is willing to accept the ghost as he is. In this way, she is very British. However, she is also quite American because she is willing to do something about the ghost, and she hopes for a better future. She is a bridge for Sir Simon, but she is also a bridge between two cultures, as evidenced by her marriage to the Duke of Cheshire.