In the books
In the A Song of Ice and Fire novels, Alys Karstark is the only daughter of Lord Rickard Karstark, Lord of Karhold, and appears much earlier in the narrative than in the TV series, before Stannis Baratheon's death and while Jon Snow is in the Night's Watch. She is described as tall and skinny, with brown hair and a long face (somewhat similar to Arya). As the Karstarks are a cadet branch of House Stark (that split off a thousand years ago) she is also a distant kinsman of the main Stark line.
Alys is a headstrong girl who knew Robb Stark and Jon Snow during childhood. Her father hoped to have her betrothed to Robb to strengthen ties with the Starks, though nothing came of it. She danced with both of them at feasts, finding Robb courteous but Jon somewhat sullen. Jon vaguely remembers that, since Alys was six years old at the time (and Jon was around seven or eight years old).
She was later betrothed to Daryn Hornwood, heir of House Hornwood, until he was killed in the Battle of the Whispering Wood. After Ser Jaime Lannister was released by Catelyn Stark, Lord Rickard promised Alys's hand to whoever brought the Kingslayer to him, in order to avenge the deaths of his sons Eddard and Torrhen, who were also killed by Jaime. Alys's surviving brother Harrion is taken captive at the Battle of Duskendale. He was imprisoned in Maidenpool, but it has been almost a year since his surviving kin received news about it, and it is uncertain if he is still alive.
Following Rickard's death, his treacherous uncle Arnolf, who secretly collaborates with the Boltons, schemes to take over Karhold: he announces in public that he supports Stannis (intending to stab him in the back while fighting the Boltons), hoping that the Lannisters will execute Harrion as a punishment, then Alys will become the heiress of Karhold. Arnolf plans to force Alys to marry his son Cregan (though he is her first cousin once removed, this is not considered incest in Westeros). They also plan to dispose of her as soon as she gives birth to a son by Cregan, who has already buried two wives.
Alys finds out about Arnolf's twofold treachery, escapes to Castle Black and informs Jon what her great-uncle is up to. She has no idea what became of her brother Harrion as a result of Arnolf's fake announcement.
Alys resents Robb Stark for killing her father. Though she knows for what crime her father was executed, it does not make a difference for her, but since Jon belongs to the Night's Watch, there is no blood feud between them, and she asks his help to save both her and Stannis from Arnolf. Jon quickly sends Tycho Nestoris to find Stannis and warns him of the traitor in his host, and he does. Arnolf, his son Arthor and grandsons are arrested, and Stannis intends to execute them: if they admit their guilt - they will be given the same swift death Robb gave Rickard; if not - they will be burned alive.
The battle of Winterfell has not come yet to a conclusion by the point the novels reached (it has not even begun), but thanks to Alys - the Boltons have been denied of one of the advantages they had over Stannis.
Meanwhile, to keep the Boltons and Arnolf from forcing Alys into a marriage to seize her birthright, Jon quickly arranges for Alys to marry Sigorn, the son of Styr and the new Magnar of the Thenns who has become one of his allies, to found House Thenn. Alys complies voluntarily, and when Cregan arrives to reclaim Alys, Jon has him arrested and placed in an ice cell.
In the television series, Alys's appearance has been delayed to after the Starks reclaiming Winterfell and the North from the Boltons. In the sixth season, Harald Karstark served as a condensation of several Karstarks, mostly Arnolf, Eddard and Alys. Harald's fate has been left unclear following the Battle of the Bastards, but given Alys's storyline and motives in the novels, it is likely Jon will betroth her to one of the wildling chieftains (it is unknown if Sigorn will appear in the series or if his role will be condensed with somebody else) to regain the support of House Karstark after executing Harald (if he is still alive).
Given that there was some condensation with the TV-Karstark family tree - loosely combining Rickard's son Harrion and his uncle Arnolf into one character, "Harald Karstark" - the familial relationship of TV-Alys is as-yet unclear: whether she will still be Rickard's daughter (and thus Harald's sister), or perhaps Harald's daughter, or a cousin, etc.
|Lord:||Harald Karstark (uncertain)||Heir:||Alys Karstark|
|Title(s):||Lord of Karhold|
|Ancestors:||Brandon the Builder · Rodrik Stark · Karlon Stark · Beric Karstark · Mara Karstark · Commard Karstark · Regan Karstark · Berrican Karstark · Petyr Karstark · Laurence Karstark · Jon Karstark|
|Current members:||Alys Karstark|
|Deceased members:||Rickard Karstark · Torrhen Karstark · Harrion Karstark|