In the most superficial analysis, Hedda is frustrated because her marriage has turned out badly. Her husband is a seemingly benign and pleasant man, but he is also remote and nerdish in an exaggerated way that would drive most spouses insane. He obviously cares more about his own intellectual work than about his wife, and he's also unrealistically self-effacing, especially when his chief concern becomes of one attempting to re-create his academic rival (also wife's ex-lover) Eilert's "brilliant" manuscript.
A deeper problem is the position of women in nineteenth-century society. Hedda clearly is a woman seeking something, clearly dissatisfied with the housewife role, even the comfortable upper-bourgeois one she has. The submissive (or even masochistic) Thea, with her unrealistic devotion to Eilert, is the opposite of Hedda, who obviously is contemptuous of her. Hedda's destruction of Eilert's manuscript is partly due to her envy of Thea's relationship to Eilert (it was "their" work, Thea's and Eilert's), but it can also be seen as an effort to help her own ineffectual husband Tesman by nullifying his rival's work, doing something actually practical even if wrong. Or it is simply a purely destructive act by which Hedda is showing her general contempt for all these people around her whom she probably regards as false and hypocritical.
This brings up a deeper issue, one of Hedda's fundamental alienation from society. It may not even be a specifically feminist issue so much as a broader existential one. Life in its normal trappings seems inadequate for Hedda, empty. The Judge's blackmailing of her, which he anticipates with glee, is merely the immediate, not the deeper cause of her suicide.