How can I analyze the character of Gurov in "The Lady with the Pet Dog"? What should I include?
The author Anton Chekhov gives you most of what you need to write a character analysis of Gurov in the second and third paragraphs of the story. If I were you I would focus on what Chekhov tells you about Gurov in the opening and add a few deductions or extrapolations. These are the pertinent paragraphs.
He was under forty, but he had a daughter already twelve years old, and two sons at school. He had been married young, when he was a student in his second year, and by now his wife seemed half as old again as he. She was a tall, erect woman with dark eyebrows, staid and dignified, and, as she said of herself, intellectual....He was afraid of her, and did not like to be at home. He had begun being unfaithful to her long ago -- had been unfaithful to her often, and, probably on that account, almost always spoke ill of women, and when they were talked about in his presence, used to call them "the lower race."
It seemed to him that he had been so schooled by bitter experience that he might call them what he liked, and yet he could not get on for two days together without "the lower race." In the society of men he was bored and not himself, with them he was cold and uncommunicative; but when he was in the company of women he felt free, and knew what to say to them and how to behave; and he was at ease with them even when he was silent. In his appearance, in his character, in his whole nature, there was something attractive and elusive which allured women and disposed them in his favour; he knew that, and some force seemed to draw him, too, to them.
Although "The Lady with the Pet Dog" is a long story, Chekhov does not continue to characterize Gurov throughout. However, as a result of his affair with Anna at Yalta, Gurov changes. He falls in love, perhaps for the first time in his life. This change of character is probably the most important thing in the story. He was just having another of his customary amorous interludes with women, but this time something entirely unexpected happens because of the influence of the woman. So you would naturally want to write about how his character at the end of the story was different from what it had been in the beginning. In other words, you would probably draw your character sketch from the information in the two paragraphs quoted above, and then explain how he had changed at the end as a result of his liaison with Anna Sergeyevna, both at Yalta and then later when they met clandestinely on rare occasions.
These two people who met by accident have had strong effects on each other. Anna has become unfaithful and deceptive. She has fallen in love with Gurov. She must like his sophistication and his daring, among other things. She must see something in him that he doesn't see in himself. This is alluded to in the second paragraph quoted above:
In his appearance, in his character, in his whole nature, there was something attractive and elusive which allured women and disposed them in his favour....
The story is told from Gurov's point of view. It is Gurov's story. Ironically, he changes from a superficial womanizer into a better, more sincere man as a result of engaging in an adulterous affair with a stranger.
I think you only need to focus on the character traits that have an influence on his relationship with Anna Sergeyevna. A character in a work of fiction is not a real human being but just an illusion of a human being. You don't need to delve too deeply into his biography or his psychology. He is just a rather ordinary middle-class man who cheats on his wife and has little respect for women until he happens to get involved with one woman who brings out feelings he had never experienced before--including, especially, the feeling of genuine love.
The story ends without any solution. Gurov has to pay for his many sins by falling in love too late in life. He is married and has children. She is married and has children. Divorce in those times was virtually impossible. They live many hundreds of miles apart. There is no solution to their mutual problem. They can't possibly be together but can only see each other on rare occasions. This unresolved ending is typical of Chekhov's stories, and he has influenced many contemporary writers, including Raymond Carver and Ann Beattie.