What is Gregor's role in the family, why does he have so much responsibility, and how does this role change?
As gpane's very clear and thorough answer shows, Gregor brings in the only income in the single-income family, supporting his mother, father and sister.
The pressure on him suggests the family's lack of resources and, perhaps, its lack of resourcefulness. Should we see the Samsa family as lacking imagination or should we see them as a family that would like to shirk work? Should we see them as a family adhering to social forms and so unwilling to entertain the notion that anyone but the lone son should be a part of the labor force?
There is an implication that the family is concerned with perceived social forms in addition to its practical concerns.
"...what mainly prevented the family from moving was their complete hopelessness and the thought that they had been struck by a misfortune as none of their relatives and acquaintances had ever been hit."
Gregor's disturbing transformation is socially awkward, of course, but its greatest impact is a negative shift in the family's financial situation. This fact is arguably the most prominent element of the story and informs Gregor's emotional situation to a considerable degree.
The family's willingness to rely on Gregor - then turn on him - suggests a willingness to also exploit him. However we characterize the family's mentality, the fact remains that when we meet them they rely on an implied assertion - only Gregor should work.
The firm he works for appears as demanding as the family.
"What a fate: to be condemned to work for a firm where the slightest negligence at once gave rise to the gravest suspicion!"
Gregor's value as a person, it seems, is derived entirely from his willingness to submit to toil. Taken for granted as a wage-earner and as an employee, Gregor is just a "bug" in the system, as it were, identified with a function and not attributed any qualities of humanity. While we may certainly want to be more sympathetic to the family in reading the story, there is ample evidence to suggest that the commercial and social expectations of the family lack a sense of humanity and instead focus on the perfunctory and the superficial.
As the previous answer makes clear, Gregor is the breadwinner of the family, and so the pressures on him are enormous. We see this at the very beginning of the story when, even after his inexplicable transformation his main concern still seems to be about his work. He muses about how much he hates his job yet he frets that he can't get there on time and is worried about his probable dismissal.
We can gauge right away, then, from his reaction to his metamorphosis, when he continues to focus on his work rather than on the startling fact that he is now a giant bug, how much he struggles under the weight of his responsibility. The family are dependent on him and the job itself seems to exert a great pressure on him. It seems that neither in his family nor at work does he have any meaningful contact whereby he can express what he really feels. It has been observed that his bizarre transformation perhaps occurs as a result of a subconscious desire to be rid of his responsibilities. If he is a giant bug, obviously he can no longer be expected to go to work.
After his change, from having a central role in the family, Gregor ends up with no role at all - unless it is to be a source of amazement, disgust and ridicule. His father, for whom he has done so much, appears unsympathetic from the start. It is only his sister Grete that actively helps him for a while, venturing into the room in which he has been secluded, feeding him and so on. But even she ends up turning against him. It is after this final rejection that Gregor decides it'd be better for all concerned if he were to die. He does just that, and subsequently the family appear to have a sense of great release and relief; they go on an excursion in the country and Grete blossoms out into a beautiful young woman.
It seems, then, that Gregor was much put-upon and personally not much valued by his family, although they relied so much on his duties as the breadwinner, and that he himself was fed up having to do so much for them. His death appears to be a great release for both his family and himself. He is finally freed from his burdensome role as the central, yet under-appreciated cog in the family wheel.
Gregor works as a traveling salesman—a situation he dislikes. He wants to quit his job, but the pressure to support his family and pay off his bankrupt father’s business debts keeps him trapped in his career. Gregor’s boss holds the father’s debts, so his job reinforces his sense of obligation to his family. Schuld, the German word for debt (the original was written in German), also means guilt in German. The symbolism of its double meaning has not been lost on many critics. Both family debts and family guilt force Gregor into intolerable employment.
Gregor never wonders why or how he has been changed from a young man into an insect. Although he worries about a great many other things, he accepts his new situation absolutely. This narrative detail is surely part of what gives Kafka’s tale its uniquely brooding mystery. For some unstated reason, Gregor acknowledges the inevitability of his fate. Establishing why Gregor and his family so naturally accept his transformation into a vile creature is central to any interpretation of the story.