How does his point of view influence the way the story is told?
Using the lawyer to tell the story makes it more difficult so that the story can be interpreted in a number of ways. The lawyer admits he is a man of "assumptions" and that he's prejudiced, not always allowing him to give an accurate view of the situation. When the lawyer tells us about Bartleby, we have to decide for ourselves whether what the narrator says is true or colored by his own prejudices. Because the lawyer is unreliable, it's more difficult for the reader to unlock the mystery of Bartleby's behavior.
The lawyer is an unreliable narrator. The reader is not getting an unbiased account of events, he is getting the lawyer's version, which includes evasions, personal prejudices, and biased reporting.