The speckled band in this story is a snake.  Specifically, it is a snake that is described as a "swamp adder."  It is supposedly the "deadliest snake in India."  It kills both Julia Stoner and Dr. Roylott.

I assume that the second part of your question refers to the death of Roylott and not to the death of Julia Stoner.  I don't think Roylott's death is properly called a murder, but Holmes has nothing to do with Stoner's death...

As to Roylott's death, Holmes causes it.  When the snake comes through the grate, Holmes strikes at it and it returns to where it came from.  When it gets there, it kills Roylott by biting him.

In "The Adventure of the Speckled Band" by Sir Arthur Conan Doyle, what is the speckled band? What role does it play in the story? In what way is Sherlock Holmes responsible for the murder in the story?

1. The speckled band pertains to the stripes, or bands, on the extremely poisonous snake from India, a swamp adder. Its owner, Sir Grimesby Roylott, killed his wife with this snake which he sent through the ventilator. Before this discovery, however, Holmes and Watson believe the eccentric Roylott may have some odd musicians around since he harbors on his property exotic animals such as a cheetah and a baboon.

2. The snake is essentially "the murder weapon."

3. Sherlock Holmes has nothing to do with the murder of Roylott's wife or of Helen's sister Julia; for these murders, Dr. Roylott is responsible. But, when he tries to kill his daughter Helen so that the wife's money will not go to anyone but him, Holmes becomes involved as Helen has hired him after her twin sister is killed.

At first Sherlock Holmes is on "an entirely wrong scent" because of the idea of "the presence of the gipsies." But when he discovers that a bell-rope that hangs down to a bed that is bolted to the floor is a "dummy," Holmes becomes suspicious and thinks of a snake since Roylott has been furnished with creatures from India. Further, a snake could deliver a poison so rapidly that the victim would instantly  be killed. And, the two little punctures which the snake would make would be barely visible. This snake could be put through the ventilator. 

All of these conditions Holmes realizes as he stands in the dark room when Helen is supposed to be sleeping. Hearing the snake hiss, Holmes instantly lights the light and attacks it, "with the result of driving it through the ventilator."  After Roylott dies, Holmes comments,

"In this way I am no doubt indirectly responsible for Dr. Grimesby Roylott's death, and I cannot say that it is likely to weigh very heavily upon my conscience."

In "The Adventure of the Speckled Band" by Sir Arthur Conan Doyle, what is the speckled band? What role does it play in the story? In what way is Sherlock Holmes responsible for the murder in the story?

As it turns out, the speckled band is a snake.  But we do not find that out until the very end of the story.  For much of the story, people suppose that it is something to do with Gypsies who live around the place of the murder.

The speckled band plays a major role in the story.  Much of the mystery centers around what in the world the speckled band could be.  Then when we find out what it is and we know that they speckled band was what killed Julia.

The snake kills Roylott.  You can say it is Sherlock's responsibility because he hits at it with his stick and drives it back into the other room where it bits and kills Roylott.