This realization happens to Bella in stages, actually. Bella is instantly drawn to Edward and singles him out as being different from the average guy, but it is Jacob who plants the seeds of thought to consider Edward as a vampire. In Chapter 6, Jacob passes off his information as slightly romanticized old tales, but the detail he conveys captivates Bella's attention. He tells her,

There's always a risk for humans to be around the cold ones, even if they're civilized like this clan was. You never know when they might get too hungry to resist . . . They claimed that they didn't hunt humans.

He goes on to explain to Bella that Edward's family belongs to this same clan of "cold ones." Bella tells him that he's a good storyteller, but then she goes home to do some internet research.

In Chapter 7, she finds a particularly striking entry online about the history of vampires:

Stregoni benefici: An Italian vampire, said to be on the side of goodness, and a mortal enemy of all evil empires.

She begins cataloging all that she reads online with her observations about Edward: "speed, strength, beauty, pale skin, eyes that shift color."

When Edward saves her life one more time in chapter eight, Bella becomes convinced that Edward can read minds. She finally decides to confront him with her body of knowledge in chapter nine, telling him, "It doesn't matter to me what you are."

Edward's voice becomes mocking, and he asks her, "You don't care if I'm a monster? If I'm not human?"

Bella replies with one simple word: "No."

It is at this point that Bella's convictions are clear. She is positive that Edward is a vampire, but she doesn't care because she has already fallen in love with him.