You will probably get many opinions on this. I used to believe that Heathcliff was not a realistic character. I used to believe he was symbolic of the themes of evil, passion, obsession, revenge, violence and cruelty that are remarkable in this novel. I did not believe that a person like Heathcliff could exist. No one could be that twisted and tormented, no matter how dysfunctional one's early life had been. As I have gotten older, I have changed my mind about Heathcliff because I have personally encountered such tortured individuals, and I have read about them in literature and seen accounts of them in the newspapers. Plus, I have personally witnessed how abandonment in childhood can have such disasterous effects on a person's adult life.
Heathcliff is a mysterious, wild and unkempt orphan that Mr. Earnshaw finds wandering on the streets in Liverpool and brings him home to raise with his own two children, Catherine and Hindley. Mr. Earnshaw favors Heathcliff over his own son, Hindley, so Hindley tortures Heathcliff -- both when they are children and as an adult. The fact that Heathcliff has been abandoned by his parents affects him profoundly in his life. The mistreatment by Hindley affects him profoundly in his life. His rejection by Catherine, whom he believes is part of him (his soul) affects him profoundly in his life. Is it any wonder that he grows up to be a vengeful, tortured man? He never receives Catherine's nurturing, redeeming love, so he is not changed by love like some other characters in literature. If Heathcliff were living today, we would be reading about him in the news. He might be an evil spouse abuser that feels his life is out of control, so he tries to control others. He might be a serial killer. He might be a rapist. You may think that this is extrapolating a lot from the story, but I think it is very possible to do so much damage to a person's psyche as a young person that someone could grow up to be a Heathcliff.
I will be interested to see what other teachers say.