What does the character mean? What is the author Checkhov trying to tell us?
I don't understand the first part of your question. Maybe you meant to ask, "What does the story mean?" A character can only mean something if he says something in dialogue.
"The Lady with the Pet Dog" is Chekhov's most famous story. What Chekhov is trying to tell us is a matter of personal perspective. I believe his main point or message is that it is a bad mistake to enter into intimate relations with another person casually and frivolously. The man is terribly bored in a boring setting at a seaside resort where the biggest event in days is the arrival of a new visitor, a lady with a pet dog. He uses the dog to strike up an acquaintance with the woman, and this leads to an intimate relationship. He doesn't realize that he is playing with fire. He is a married man who has had many affairs in the past--but this time he falls in love. Unfortunately, both of them are married and have children. It is too late in life for them to get divorced and to marry. She falls in love with him, too. The situation is further complicated by the fact that they live hundreds of miles apart. At the end of the story it is apparent that they will both suffer for years because they cannot be together. A religious-minded person might say that God is punishing them for adultery (as Francesca da Rimini and Paolo da Rimini were punished in Dante's Inferno). But Chekhov, like Tolstoy in his Anna Karenina, seems to be saying that adultery, like other misguided behavior, carries its own punishment with it.
Many men and women enter into intimate relations thoughtlessly, casually, maybe out of boredom or intoxication, and then learn to regret it for one reason or another.