Heathcliff succeeds spectacularly (and mysteriously, and, perhaps, unrealistically) to take things from those whom he thought had wronged him -- specifically Hindley Earnshaw. While one of the "lessons" (if they can be called that) of the novel is that revenge never brings joy to those who practice it, nevertheless Heathcliff exacts very specific, reciprocal revenge on Hindley, and does it thoroughly. Hareton, Hindley's son, finally escapes Heathcliff's revenge, but by any measure Heathcliff certainly had gotten enough of it already.
Heathcliff came to the Earnshaw house as a penniless orphan. Old Mr. Earnshaw treated him with respect and love, and welcomed him into the family. But when Mr. Earnshaw died, Hindley became the head of the household, and he treated Heathcliff as a servant: worse, he ridiculed and reviled him.
Once Cathy was married to Edgar Linton, Heathcliff knew he had no reason to stay any longer at Wuthering Heights and he disappeared. He returned, years later, with a fortune (how that fortune was obtained is never explained -- Emily Bronte was largely ignorant of the male, commercial world, and she only hinted that Heathcliff's wealth was obtained illegally). What Heathcliff exacerbates Hindley's weakness for drink and gambling, and he eventually takes everything -- including Wuthering Heights -- from him. Thus he is revenged upon Hindley (and Hindley's son, Hareton, whom he treats in the same fashion as he, Heathcliff, was treated).
Heathcliff is "revenged" upon Edgar Linton, whom he perceives as having taken Cathy from him, by taking Isabella, Edgar's sister, from him. Heathcliff does not love Isabella, but he courts her away from her brother and marries her purely to spite Edgar and Cathy. Thus he is "revenged" upon them, because once he and Isabella are married he is horribly cruel to her.
The revenge plot eventually peters out with the new generation of Heathcliff and Cathy's children, but Heathcliff's desire for revenge continues until he dies. It is a defining characteristic of Heathcliff, and, for some readers, it is clear that he feels this way because he was so ill-treated himself (by Hindley, after Old Mr. Earnshaw dies). Whether you believe it was an inborn character trait of Heathcliff, or it was a product of his upbringing, is your interpretation of the novel. Ultimately, Heathcliff, like all those who desire revenge, does not get peace and happiness from it.