Heathcliff's death,which actually occurred three months ago, is reported orally by Nelly Dean to Lockwood: "three months since...I'll tell you all about it" Ch.32.
In Ch.34 Nelly gives a detailed daily account of the deterioration of Heathcliff's health and how he finally dies in April 1802. Heathcliff passes sleepless nights wandering outside the house. He refuses to take any food and wishes to be left alone. But most importantly, "he had a strange joyful glitter in his eyes that altered the aspect of his whole face." Nelly soon realises that he has seen Catherine's ghost and that shortly he will die. He refuses to accept Nelly's advice to repent of his evil past and seek seek God's forgiveness. That night he locks himself up in his room "graoaning and murmuring to himself." He refuses to see anyone even the doctor. Next day morning Nelly discovers him dead with his hand stretched out on the open window sill, hinting that he has become a ghost just like Catherine's. It is reported that the villagers have seen the ghosts of Catherine and Heathcliff.
In Ch.29 he reveals to Nelly that he had bribed the sexton to remove one side of Catherine's coffin so that his body can be placed on that side so that at least after his death he can be united with her. In Ch.34 he is buried just "as he had wished."