Family Background

We actually know quite a few Bones, so we have something to go on here.

The first one of note is Amelia Bones, Susan's aunt:

“She’s my auntie,” she said. “I’m Susan Bones. She told me about your hearing. So — is it really true? You make a stag Patronus?”

(OotP)

She is the head of the DMLE:

“Interrogators: Cornelius Oswald Fudge, Minister of Magic; Amelia Susan Bones, Head of the Department of Magical Law Enforcement; Dolores Jane Umbridge, Senior Undersecretary to the Minister. Court Scribe, Percy Ignatius Weasley — ”

(OotP)

Her position of importance at the ministry, to me, suggests the Bones family have been involved in the Wizengamot (I believe department heads more often than not are chosen from within the Wizengamot). And we know until the first war they were a larger family, so they're likely mostly pureblood, though not enough to appear in the sacred 28 (I assume that they married half-bloods and muggleborns, but all the recent ones are probably wizards).

Amelia is cold and savvy enough to hold her difficult position, which indicates a lot about her disposition. I'll note that in Harry's trial, she seems aware of what Fudge is doing and doesn't seem to dislike Harry at all:

“Yes, Professor Lupin taught me in my third year, because of the — ” “Impressive,” said Madam Bones, staring down at him, “a true Patronus at that age... very impressive indeed.” Some of the wizards and witches around her were muttering again; a few nodded, but others were frowning and shaking their heads. [...] “Ah,” said Fudge again, smirking unpleasantly as he looked around at the Wizengamot, as though inviting them to share the joke. “Yes. Yes, I thought we’d be hearing something like this.” “Dementors in Little Whinging?” Madam Bones said in tones of great surprise. “I don’t understand — ” [...] “I may be wrong,” said Dumbledore pleasantly, “but I am sure that under the Wizengamot Charter of Rights, the accused has the right to present witnesses for his or her case? Isn’t that the policy of the Department of Magical Law Enforcement, Madam Bones?” he continued, addressing the witch in the monocle. “True,” said Madam Bones. “Perfectly true.”

(OotP)

She is riding the line to keep her job while being as just as she can. She is a politician and not a bleeding heart.

She seems suspicious of Mrs. Figg's testimony, I think she knows she's lying but she lets Harry off the hook anyway because she probably knows Fudge is bullshiting in this case:

“Running?” said Madam Bones sharply. “Dementors don’t run, they glide.” “That’s what I meant to say,” said Mrs. Figg quickly, patches of pink appearing in her withered cheeks. “Gliding along the alley toward what looked like two boys.” [...] “What did they look like?” said Madam Bones, narrowing her eyes so that the monocle’s edges disappeared into her flesh. “Well, one was very large and the other one rather skinny — ” [...] “Not a very convincing witness,” said Fudge loftily. “Oh, I don’t know,” said Madam Bones in her booming voice. “She certainly described the effects of a dementor attack very accurately. And I can’t imagine why she would say they were there if they weren’t — ”

(OotP)

Susan seems close to her aunt (who feels comfortable gossiping about closed trials with her) and many of their family died in the first war:

No one lived after he decided ter kill 'em, no one except you, an' he'd killed some of the best witches and wizards of the age - the McKinnons, the Bones, the Prewetts - you were only a baby an' you lived.

(PS)

matiaskanfunfa asks: There is a girl named Susan Bones who was sorted in the first book and there was a family called the Bones that Voldemort tried to destroy, is this a coincidence or will Harry meet her in future books? jkrowling_bn: Susan Bones' grandparents were killed by Voldemort!

(Interview)

“That’s Edgar Bones ... brother of Amelia Bones, they got him and his family too, he was a great wizard ... 

(OotP)

So it seems Amelia, her brother (Susan's father) and Susan's mother are the only Bones still around.

It also shows they stood in opposition to Voldemort very clearly. In the second war, one of the first official casualties of the war is Amelia Bones who is killed by DE between OotP and HBP. So it seems the Bones family did a lot against Voldemort and that Amelia was thought of as competent and effective enough in her job to be considered a threat that needed to be eliminated. (I believe that if she was alive, she would've probably been promoted to temporary minister instead of Scrimgeour, talked about this more here)

Susan Herself

“Bones, Susan!” “HUFFLEPUFF!” shouted the hat again, and Susan scuttled off to sit next to Hannah.

(PS)

Susan was born in 1979/1980 and is in Harry's year. Her appearance isn't actually known, but in official art, she has light skin and auburn hair. She is one of the 40 students in Harry's year from JKR's very early notes where she is noted down as a half-blood. Though whether that is because both her parents are half-bloods, one is a muggle or one is a muggleborn is unknown. I feel like we would hear more about her in book 7 if one of her parents was a muggle or muggleborn, so she's probably a "half-blood" mostly because that's wizards' catch-all phrase for anyone who isn't pureblood or muggleborn.

She was sorted into Hufflepuff, probably like her aunt, Amelia, and her uncle, Edgar. It's very possible her father (Amelia's and Edgar's brother) was also a Hufflepuff as houses seem to run in families.

“And nobody helped you get rid of those dementors this summer,” said Susan Bones. “No,” said Harry, “no, okay, I know I did bits of it without help, but the point I’m trying to make is — ”

(OotP)

She is supportive of Harry and his achievements, which always makes me like a character just a little bit more, but she tells him what he needs to hear to his face (not the only one in this scene), but it's not the entire D.A. doing this, yk?

But the really interesting part about this is that Amelia apparently shares closed (?) trial proceedings and legal gossip with her niece. And Susan treats this like a regular, normal occurrence. This implies two things:

Susan and Amelia are pretty close.

Susan is interested and curious enough in these things that Amelia bothers to tell her — this implies Susan likes to hear gossip and might have an interest in wizarding politics.

who now found themselves the unwilling objects of a gruesome sort of reflected fame as they walked the corridors: Susan Bones, who had an uncle, aunt, and cousins who had all died at the hands of one of the ten, said miserably during Herbology that she now had a good idea what it felt like to be Harry. “And I don’t know how you stand it, it’s horrible,” she said bluntly, dumping far too much dragon manure on her tray of Screechsnap seedlings, causing them to wriggle and squeak in discomfort.

(OotP)

This above quote is my favorite in relation to her. It indicates a few things about her:

She is blunt and straightforward.

She understands Harry doesn't like his fame (a rare thing for characters Harry isn't close to)

She doesn't treat Harry like the great distant Boy-Who-Lived.

She is a Hufflepuff who isn't amazing at Herbology, and I love that for her. I headcanon she got, like, an A in Herbology and dropped it in 6th year.

Her grief seems to be quiet and angry, not unlike Harry's. And I love Harry.

Firstly, Malfoy, Crabbe, and Goyle, who had clearly been waiting all week for the opportunity to strike without teacher witnesses, attempted to ambush Harry halfway down the train as he made his way back from the toilet. The attack might have succeeded had it not been for the fact that they unwittingly chose to stage the attack right outside a compartment full of D.A. members, who saw what was happening through the glass and rose as one to rush to Harry’s aid. By the time Ernie Macmillan, Hannah Abbott, Susan Bones, Justin Finch-Fletchley, Anthony Goldstein, and Terry Boot had finished using a wide variety of the hexes and jinxes Harry had taught them, Malfoy, Crabbe, and Goyle resembled nothing so much as three gigantic slugs squeezed into Hogwarts uniforms as Harry, Ernie, and Justin hoisted them into the luggage rack and left them there to ooze.

(OotP)

She is part of the D.A. and like many of them defends Harry and likely fights in the battle of Hogwarts later, though her fate there is unclear I tend to assume she survived. I like to assume she survived and married someone else from the D.A.

But the above implies that like others in the D.A. she is proficient with hexes and defensive magic Harry taught them and perhaps other spells they studied up on their own.

The second attempt was no better than the first. The third was just as bad. Not until the fourth did anything exciting happen. There was a horrible screech of pain and everybody looked around, terrified, to see Susan Bones of Hufflepuff wobbling in her hoop with her left leg still standing five feet away where she had started. The Heads of House converged on her; there was a great bang and a puff of purple smoke, which cleared to reveal Susan sobbing, reunited with her leg but looking horrified.

(HBP)

She had a bad time in apparition lessons, I assume it was about the focus. I think she was too distracted and didn't apparate properly. (Like her plants' squeak because of her inattention when thinking about how annoying the attention she's getting is in an earlier quote).

So, basically, I like the potential of Susan's character as a half-blood-ish Hufflepuff, who's brave, blunt, and straightforward, interested in wizarding politics, dislikes Herbology, closed off emotionally, close to her Auntie, gets angry but is usually chill, can actually read people well, and has a tendency to get distracted by her own thoughts but is magically capable when she makes an effort.