Tennessee Williams’s life growing up was profoundly affected by his father’s absence, his mother’s authority, and his sister Rose’s illness. How did his own dysfunctional family dynamic help to develop the relationships he describes in The Glass Menagerie? How was his own family similar to the family he created in the play?

As the question addresses, Williams's family situation is nearly identical to the family situation in The Glass Menagerie. The father is absent apart from his framed portrait in the living room, an absence which causes the mother, Amanda Wingfield, to become obsessive over the security of the family. Williams's actual mother was the daughter of a preacher who treated her family to a strict, sometimes hysterical leadership. Amanda Wingfield's obsession makes her overbearing and intolerable to Tom, who is a stand-in for Williams.
Tom wants desperately to be a writer, but he is working at a shoe factory and spending his evenings drinking and carousing. This was Williams's exact life situation after his family pulled him from college and put him to work at the shoe factory where his father was working after years of being a traveling salesman. During this time of discontent, Williams grew very close to his younger sister. In actuality there was another sibling, a younger brother, but in the play there is only Tom and Laura. Laura was based on Williams's beloved sister, Rose, who suffered from mental illness and eventually underwent a lobotomy to treat her sickness.
Laura is also apparently unwell, though her illness is not explicit. She is unusually preoccupied with her glass collection—the inspiration for the title—and she is childlike and aloof at times in the play. Her love-interest, Jim O'Connor, calls her "Blue Roses," undoubtedly a reference to Williams's sister's name. Amanda wishes desperately for Laura to find a boy who will marry her and care for her, providing the security she knows she will not be able to provide forever, but Tom feels uncomfortable setting up his sister. Williams's sister never married, and we are led to believe that Laura never will either, as she is so devastated by Jim's engagement to another girl.
Her devastation, however, is quiet, and Tom reacts to the situation by fleeing—choosing his own life over those of his mother and sister, who he feels unable to care for or tolerate emotionally. Williams also left his family eventually, moving around the US, drinking and writing, until he got The Glass Menagerie produced in Chicago. The play went on to be performed on Broadway, and from there Williams achieved the success he'd dreamed of when he was a lonely factory worker like Tom.