It's All About Me: It's debatable whether he is an example of this or not, since he doesn't really care if most others are happy, but still very much gravitates around people. When he cannot center his life around being with Cathy, he starts obsessing over getting revenge from everyone.
Lack of Empathy: He very often expresses his bewilderment at people who imagine that he will adopt a perfectly understandable and legitimate behavior towards them.
Mysterious Past: For all of Heathcliff's life that we do know, he's still made of this trope. We don't know anything about his early years, to age seven or so, or why he couldn't speak English when he first came to the Heights or what his name might have been before that time. The mystery only deepens in the three years he spends away from the Heights and somehow has made himself so rich in that time that he's bought the house from under Hindley's nose.
Scary Black Man: The 1991 edition's illustrations depict him as one, towering over an intimidated Hareton and Cathy Junior. Whilst the novel explicitly points out that he's not 'a regular black', it does repeatedly mention how sinister and intimidating his dark skin, hair, and eyes make him look.
All Girls Want Bad Boys: Deconstructed - the love between Catherine and Heathcliff is passionate, but it is also clearly unhealthy and intensely destructive, leading to nothing but the ruin of the lovers and almost everyone around them.
Genki Girl: At least until the residents of Thrushcross Grange get hold of her.
Hysterical Woman: Driven to this towards the middle of the novel, going as far as searching for Heathcliff across the moors during a storm this causes her to catch a fever and die
It's All About Me: Well, since Heathcliff is the only one she has any deep-seated interest for and she basically says she is him (and probably wants to identify entirely with him; see below), she may be a literal example. She is initially as abusive as her brother is towards Heathcliff, then manipulates Linton, doesn't care about the health of her brother, her sister-in-law, or poor Nelly, to whom she is very much a Jerk with a Heart of Jerk.
Love Makes You Crazy: Her love for Heathcliff is passionate but incredibly unhealthy and twisted, and eventually consumes her with insanity. Pauline Nestor's introduction to the novel argues that her desire for total identification with the object of her love ('I am Heathcliff!' rather than 'I love Heathcliff') represents a regression to childlike lack of identity, also shown by her failing to recognise her own reflection in a mirror just before she dies. Catherine's love is so insane that it basically destroys her entire identity and personality.
Nice Job Breaking It, Hero: Her desire to have her cake and eat it (marrying Edgar for money and status while keeping Heathcliff on the side) ultimately leads to her own death and Heathcliff's descent into madness and need for revenge.
Asshole Victim: If you subscribe to the theory that Hindley is murdered by Heathcliff.
Dramatic Irony: Heathcliff's later treatment of his son Hareton mirrors the way Hindley himself treats Heathcliff when his father dies and he inherits the estate.
Dropped a Bridge on Him: Hindley's death occurs off-page but is assumed to be a result of his drunkenness. Only Heathcliff is said to be with him when he dies and it is implied that he gambles away ownership of Wuthering Heights to the latter during this period.
Drowning My Sorrows: After his wife's death in childbirth. Its also strongly implied that Heathcliff encourages this.
Badass Bookworm: Despite coming across as a nerd and a weakling, thrashes Heathcliff the one time they actually fight. Forever after, Heathcliff won't risk confronting him unarmed, even during the many long, solitary walks Edgar takes out on moors.
Genre Savvy: He seems to anticipate Cathy and Hareton's later relationship.
Horrible Judge of Character: Played with; he's inclined to think of Heathcliff as a "capital fellow" because he wants his landlord to be antisocial and reclusive. After hearing Nellie's full story, he seems to have a highly critical opinion of Heathcliff, but he doesn't do anything about it as he owes him six-month's rent.
Tellingly, he also has a very high (and accurate) opinion of Hareton Earnshaw, when everyone else considers him an uncultured lout, because he notices Hareton being torn between loyalty to Heathcliff and Cathy.
Jerk with a Heart of Gold: Unlike her mother, who is this at her best, she is this at her worst, at the beginning of her storyline and during her Broken Bird phase, when she asks for Lockwood to be lead back home, instead of let out to get lost in the swamps under the tempest.
Spoiled Sweet: Even has some snippets of this before her Break the Haughty storyline, like when she starts taking care of Nellie when the latter is sick and she sacrifices nights out with Linton for this. Only after the aforementioned Break the Haughty does she switch to this as her default attitude.
Strong Family Resemblance Nellie notes that although she looks nothing like Catherine except for having the same dark eyes, they are very similar in personality, particularly in their spirited and stubborn natures, although the younger Catherine's upbringing by Nellie and Edgar seems to temper her wilder impulses.
Daddy Issues: Oh, boy... Who wouldn't have issues with a father like his ?
Dirty Coward: This is how he manages to mic his parents's worst traits, and his apparent motviation for obeying Heathcliff, but there is actually more to it. His behavior seems to be partly motivated by a genuine pleasure taken in hurting others, or seeing Heathcliff hurt them.
Anti-Villain: He seems to be a type II-III, constantly undergoing trauma, first because of his father, and later Heathcliff, and believing the latter to have at least some right to do things as he does... He later pulls a Heel-Face Turn for Cathy.