Heathcliff outlives his beloved Catherine by several decades. Throughout that time he is angry and tortured by the loss. He is determined to destroy the families he feels have destroyed Catherine and him, and he lashes out at these families repeatedly and cruelly. He is an anguished man on a mission of revenge.
We witness, but don't understand, his agony the night Lockwood sleeps in Cathy's old room at Wuthering Heights early in the novel. Heathcliff becomes distraught when he hears of Cathy's ghost knocking at the window to come in, and of Lockwood's refusal to allow her entry.
But at the end of the novel—a time only a few months after Lockwood spends the night—Heathcliff suddenly changes. He's no longer in a frenzy of anger and despair. He watches the young Cathy and Hareton coming together as young lovers, as he and the first Cathy once did. It puts him into a certain degree of agony to watch them, but he no longer has the desire to destroy them. He could enact revenge, but his will for it has gone.
Nelly, there is a strange change approaching; I’m in its shadow at present. I take so little interest in my daily life that I hardly remember to eat and drink.
It becomes apparent that he is in communication with Catherine's ghost and that it is beckoning him to her. He suffers to be still alive on the earth, and he tells Nelly that his paradise—being with Cathy—is close at hand. He makes his will, and he leaves instructions with Nelly about his burial. Finally one morning, Nelly enters his room to find:
Mr. Heathcliff was there—laid on his back. His eyes met mine so keen and fierce, I started; and then he seemed to smile. I could not think him dead: but his face and throat were washed with rain; the bed-clothes dripped, and he was perfectly still. The lattice, flapping to and fro, had grazed one hand that rested on the sill; no blood trickled from the broken skin, and when I put my fingers to it, I could doubt no more: he was dead and stark!