One of the simplest ways Holmes displays this character trait is through direct address. When Helen is first telling her story, he asks, “Pray be precise as to details.”
Later, once the whistling sound has been described, Holmes says the following:
“One moment,” said Holmes, “are you sure about this whistle and metallic sound? Could you swear to it?”
Here he is checking her story, making sure as to specifics. When he actually visits the house, he walks slowly back and forth from the outside, simply observing the house, to see if there are any signs of oddity.
All of these are signs of Holmes' precise and meticulous nature. That he is responsible as well is shown in the apparent guilt he feels over the story's ending:
" 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.”