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.
Heathcliff wants to revenge himself on the people he holds responsible for keeping him apart from Catherine: Hindley Earnshaw and Edgar Linton. He overhears a part of a conversation that Catherine has with Nelly Dean about her decision to marry Edgar. When Nelly asks if she loves Edgar, Catherine says yes, and she also says she couldn't possibly marry Heathcliff now because of the way Hindley has degraded him to the status of farm hand. Heathcliff runs off before he hears the part of Catherine's speech in which she talks about how deeply she loves Heathcliff, says she knows her love for Edgar is much more shallow, and states she is marrying Edgar primarily to use his money and influence to help Heathcliff.
Heathcliff goes away for three years and returns a wealthy gentleman. He wants revenge on Hindley for having mistreated him. Heathcliff was adopted by the old Mr. Earnshaw to be a son of the family, but after his death, Hindley's jealousy of Heathcliff, along with his grief and alcoholism over having lost his young wife, causes him to ruin Heathcliff's life. Heathcliff seeks and gains revenge from him by winning Wuthering Heights through gambling. Further, he becomes the guardian of his young son, whom he mistreats.
Heathcliff wants revenge on Edgar for daring to marry Cathy. He holds Edgar responsible for Catherine's death, believing he is too much of a weakling to understand her. Heathcliff also blames Edgar for her death by separating Catherine from him and banning Heathcliff from his house. He seeks revenge on Edgar by marrying and abusing his sister, Isabella, and by manipulating Edgar's daughter into marrying his son with Isabella.
Heathcliff is a character who is tormented by his love for Catherine and by his bitterness over his ill treatment by Hindley. Heathcliff has been treated as a favored son by Mr. Earnshaw, and then after Mr. Earnshaw's death, Hindley exacts revenge on Heathcliff by treating him as a base servant. Heathcliff feels further rage and bitterness when he overhears Catherine tell Nellie that she would be "degraded" by loving Heathcliff. He runs away, filled with hatred for all who have hurt him. When he comes back to Wuthering Heights, he comes back as master of the house, prepared to get revenge on his enemies. After Catherine's death, his bitterness intensifies, causing him to act vindictively and cruelly to the younger generation.
The root of Heathcliff's motivation to take revenge is unrequited love. Though there is great suggestion at the start of the novel that Catherine loves Heathcliff, she can't allow herself to marry below her station. Though she cares for Heathcliff and is loving, she can't offer him the deep sort of love which he desires. Other factors only exacerbate this feeling. Catherine's marriage to Linton, a wealthy neighbor in the countryside, and Catherine's brother who treats Heathcliff very poorly. Upon Heathcliff's return later in life, the other characters are surprised to see that he has become a gentleman. He has returned to buy Wuthering Heights and inflict revenge on those who separated him from Catherine. The separation from Catherine is what motivates him to inflict revenge on the others.