For someone as obscure as Daphne Greengrass is canonically, she sure gets a lot of attention in fanfic. We can reasonably assume she's a pureblood (as her sister marries Draco), and she most likely got sorted into Slytherin (though a handful of fics do put her in Gryffindor or Ravenclaw).

Because she's effectively a fresh block of moldable clay for fanfiction writers, she can be interpreted many different ways. Some pieces have her as nasty as the Wicked Witch of the West; others depict her as almost sickeningly sweet. Many pieces also have her as an "ice duchess/princess/queen" where she slowly falls in love with Harry.

I personally see Daphne as a very quiet and studious girl, not unlike Hermione, but she's more aloof and truly values her privacy. Although she's a Slytherin and does have plenty of ambition and a dash of cunning, at her core she knows that blood supremacy is wrong, but has difficulty finding the courage to stand up against Draco and his goons.

She has few friends, but much of that is by choice (again, she takes her studies seriously, and gets rather testy if interrupted) and some of her housemates, particularly the boys, give her the unflattering nickname of "Ice Princess" (or Duchess or Queen, whatever) as she doesn't give them the time of day. Her closest companion is Tracey Davis, and she has an amicable (but not close) relationship with Pansy Parkinson and Millie Bulstrode.

She is intrigued by Harry and his companions, perhaps even sympathetic to a degree. She doesn't socialize with them much, particularly during their earlier years, but I would imagine by their fourth year she might have a brief, casual conversation with Harry or perhaps Hermione (perhaps studying the same subject in the library would be a good conversation starter). They gradually discern that she's not like many of her housemates - she may be ambitious with a dash of cunning, but she still plays by the rules and truly possesses a kind heart. However, she is wary of a full-on friendship as being seen in public with Harry or his sympathizers would be viewed as blasphemous in Slytherin. In her heart of hearts, she knows Harry and his friends are much better company than her Death Eaters-in-training housemates.

She can also be rather snarky and sassy, often making wisecracks at her classmates' expense, but she's also got a good amount of sweetness to balance out the sarcasm. For example, she might privately console a sobbing first or second-year Muggle-born student that might have been maliciously teased by some of her housemates.