At its heart, Heathcliff's revenge is about his forced separation from Catherine. The two grew up as adopted siblings and were incredibly close. However, once Catherine becomes friends with the Lintons, she begins to spend less time with Heathcliff. Her social ambitions grow, and even though she still loves Heathcliff more than anyone else, she marries Edgar Linton in order to become a fine lady. Not realizing how much Catherine loves him, Heathcliff runs away, gaining a fortune in the process. When he returns, he puts his revenge plan into action. He charms and marries Isabella Linton, estranging her from her family and then subjecting her to domestic abuse. Years later, he forces Edgar and Catherine's daughter to marry his son, giving him possession of the Linton house, Thrushcross Grange.
It must also be noted that Heathcliff also desires revenge on Hindley, who abused and exploited him after the death of Mr. Earnshaw. Hindley forced Heathcliff to become a servant in his own house and tried keeping him apart from Catherine even as he drank his family into ruin. When Heathcliff comes back wealthy, he buys up Wuthering Heights and forces Hindley into further degradation, encouraging his drinking habit until he finally dies. He also makes Hindley's young son Hareton a servant.
So, Heathcliff's vengeance is about getting back at the people who degraded him and kept him away from Catherine. Ultimately, the vengeance neither fulfills Heathcliff nor entirely destroys everyone he targets. Hareton and Catherine Linton push back against Heathcliff's hatred, ending the cycle of abuse he chose to continue. As a result, Heathcliff ends up wasting away, unsatisfied with his revenge and longing to reunite with Catherine in death.