Miss Murdstone is the unmarried elder sister of Mr. Murdstone. As soon as Mr. Murdstone moves in to Mrs. Copperfield's house, Miss Murdstone accompanies him. And later on, when David hears of Mr. Murdstone's remarriage, it appears that Miss Murdstone has followed him there, too. We don't get much psychological depth to explain Miss Murdstone's cruelty. She really, really just seems to loathe David. She does her best to encourage Mr. Murdstone to beat David, and she frequently sneers and criticizes both David and poor Mrs. Copperfield.
Miss Murdstone is a woman without much power and influence of her own. We could speculate that she is as sadistic as Mr. Murdstone is but lacks the authority to bully people on her own. So, she needs to accompany Mr. Murdstone, and to be cruel vicariously through him. Which doesn't seem like much of a life.
A useful plot device, Miss Murdstone helps provide obstacles to David's marrying Dora. Out of the blue, Miss Murdstone takes a temporary job in Mr. Spenlow's household as a companion and chaperone for Dora once her mother dies. The mechanics of why she does this are unclear, but Dickens effectively illustrates the mutual surprise and disgust when she and David meet again by accident after a good seven or eight years of separation. Miss Murdstone suggests to David that they both let the past lie: they shouldn't reveal to the Spenlows how they know each other.
David agrees, but it's not so simple as that: Miss Murdstone still apparently hates David's guts and wants to damage him in any way that she can. So, when Miss Murdstone finds secret letters between David and Dora arranging their dates, Miss Murdstone immediately takes this evidence of a secret romance to Mr. Spenlow, without listening to Dora's pleas or protests. And it's on the evidence of these letters that Mr. Spenlow forbids David from getting engaged to Dora. Of course, Mr. Spenlow dies suddenly in, like, the next chapter, so all of that suspense immediately gets squashed. And Miss Murdstone vanishes, never to reappear in the novel again except as an object of gossip in relation to her brother's second marriage.