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.