How did Julia Stoner die?

Short Answer: Julia dies after being bitten by her step-father's venomous swamp adder.
At the beginning of the story, Helen Stoner visits Sherlock Holmes and Dr. Watson and explains to them that she fears for her life. Helen goes on to mention that she views her step-father, Sir Grimesby Roylott, as a threat to her well-being and explains to Holmes and Watson that her sister, Julia Stoner, died two years ago. Helen says that Julia died shortly after she got engaged and the events surrounding her mysterious death are eerily similar to her current situation. According to Helen, on the night of Julia's death, her sister mentioned that she had heard a mysterious low whistle for the past three nights. Later that night, Helen was awakened by Julia's screams and happened to hear a low whistle whenever she opened her door. Before Julia died in Helen's arms, Julia told her,

"Oh, my God! Helen! It was the band! The speckled band!" (Conan Doyle, 4).

At the end of the story, it is discovered that Julia was bitten by a venomous swamp adder, which Sir Grimesby Roylott trained to travel between his room and Julia's room whenever he blew a low whistle. Eventually, Julia accidentally provoked the venomous snake, which caused it to bite and kill her.
After Helen got engaged, construction started on her home and she was forced to stay in Julia's former room. She also heard the mysterious whistle, which motivated her to seek Holmes's services. It is discovered that her step-father also planned to kill her using the same method, but his plan backfires after Holmes intervenes.

Julia Stoner died in agony two years before her twin sister Helen came to London to ask Sherlock Holmes for advice and assistance. We do not learn the cause of Julia’s death until towards the end of the story. The author, Sir Arthur Conan Doyle, never offers an exact description of how Julia’s death was caused by her wicked stepfather. This was mainly because the notorious prudery of Victorian times prevented the author from depicting a young woman in her nightgown in her bed. Actually her own sister didn't see her until she staggered out into the corridor and died in her arms. But Conan Doyle undoubtedly intended to have the reader imagine exactly what had occurred in that bedroom. What was not told to the reader in so many words was in fact more horrible than anything explicitly described. Here is what must have happened.
For several nights running Dr. Roylott slipped his poisonous snake through the ventilator. It slithered down the dummy bell-rope onto the pillow beside the sleeping girl’s head. Why didn’t it try to escape from its cruel captivity now that it was at least partially free? The snake comes from a tropical climate and the weather is remarkably cold, as the author suggests many times.
"I am glad to see that Mrs. Hudson has had the good sense to light the fire. Pray draw up to it, and I shall order you a cup of hot coffee, for I observe that you are shivering.”
“It is a little cold for the time of the year,” said Holmes.
The central portion was in little better repair, but the right-hand block was comparatively modern, and the blinds in the windows, with the blue smoke curling up from the chimneys, showed that this was where the family resided.
The tropical reptile's natural inclination would be to seek warmth. Dr. Roylott must have known that. On each of the nights the snake crawled down the bell-rope it would not only have remained on the bed, but it would have crawled under the bed-covers and curled up right beside the sleeping Julia. The snake had no reason to attack the girl whose body heat was keeping it warm, but sooner or later, as the murderer knew, the girl would turn over in bed and right on top of the swamp adder. When that happened, the snake would bite her through her nightgown. She would scream, and Roylott would immediately blow his whistle to summon the “speckled band” back up the bell-rope and through the ventilator, where he could capture it and return it to the steel safe.
The reader—and especially the female reader—can imagine how horrible it would be to have a snake as a bed companion. The same thing would have happened to Helen if her sister hadn’t told her about that mysterious whistle. Helen heard it on only one occasion and went to see Sherlock Holmes immediately next morning—but she must have had the swamp adder in bed with her for at least one night.