Cathy Earnshaw was once a headstrong child, and that is undeniable. When we analyze her at age six, we see a child with a clear knowledge of her background and rank. She is especially aware that she is above-status when compared to Heathcliff.
As she becomes older, we see a somewhat conceited woman whose headstrong nature makes her overlook her emotions. She ends up marrying a man that she does not love simply because he is of the same class as her own. She leaves Heathcliff alone and abandoned, not knowing that the wrath that is coming her way would end up killing her.
However, it is precisely when Heathcliff takes control of his life that Cathy becomes aware of what she lost. This is the moment when she becomes a tragic hero. No longer do we see bouts of impertinence coming from her, nor the spitefulness that she once was capable of feeling for those below her. Now she sees what her life has become, and that life can change completely. Slowly, she becomes a victim of Heathcliff. She continues to love him, long for him, and desire him. Yet, she also wonders about him and, what is worse, she fears him completely.
When she dies in childbirth there is no question in the reader's mind that she is still thinking about Heathcliff. Therefore, it is arguable that Cathy was meant to be a spoiled brat due to her status in society. However, that changed when she faced reality and saw the new Heathcliff coming back into her life.
Bronte certainly provides quite a bit of evidence to support an interpretation of Catherine as being headstrong and selfish. As a child, she is determined to get her way and likes to control others (consider her request of a whip as a present). Similarly, her choices (to continue to love Heathcliff and to marry Edgar for selfish reasons) demonstrate that she is self-centered and manipulative. Indeed, it is difficult to feel much sympathy for Cathy.
However, Cathy's flaws actually contribute to an accurate description of her as a tragic romantic heroine. Like all tragic heroes/heroines, Catherine Earnshaw falls from a place of prominence and luxury to an early death and discontent. Also--like all tragic heroes--Cathy's fall results from her own tragic flaw (selfishness and misplaced "love"). While Heathcliff's manipulation of almost every character in the novel contributes to Cathy's early demise, Bronte presents her as an independent enough character who could have escaped Heathcliff's wiles if she had desired to do so.
Thus, while Catherine Earnshaw does not inspire as much sympathy from readers as do other Victorian tragic heroines such as Hardy's Tess, she stills demonstrates most of the attributes of a tragic hero/heroine.