Why did Umbridge hate Muggles/half-breeds, when she’s not a pure-blood? Weren’t her mother and brother Muggles? And why did she hate Harry so much?
Dolores just didn't have a great start in life and it all went downhill from there. As we read on Wizarding World (formerly Pottermore).
You are correct, Dolores was a half-blood. A half-blood who resented both halves.
Dolores Jane Umbridge was the eldest child and only daughter of Orford Umbridge, a wizard, and Ellen Cracknell, a Muggle, who also had a Squib son. Dolores’s parents were unhappily married, and Dolores secretly despised both of them: Orford for his lack of ambition (he had never been promoted, and worked in the Department of Magical Maintenance at the Ministry of Magic), and her mother, Ellen, for her flightiness, untidiness, and Muggle lineage. Both Orford and his daughter blamed Ellen for Dolores’s brother’s lack of magical ability, with the result that when Dolores was fifteen, the family split down the middle, Orford and Dolores remaining together, and Ellen vanishing back into the Muggle world with her son. Dolores never saw her mother or brother again, never spoke of either of them, and henceforth pretended to all she met that she was a pure-blood.
This attitude prevailed through her life as a witch and her work in the Ministry denying her lineage and estranging her father, doing anything she could to be on top.
An accomplished witch, Dolores joined the Ministry of Magic directly after she left Hogwarts, taking a job as a lowly intern in the Improper Use of Magic Office. Even at seventeen, Dolores was judgemental, prejudiced and sadistic, although her conscientious attitude, her saccharine manner towards her superiors, and the ruthlessness and stealth with which she took credit for other people’s work soon gained her advancement. Before she was thirty, Dolores had been promoted to Head of the office, and it was but a short step from there to ever more senior positions in the management of the Department of Magical Law Enforcement. By this time, she had persuaded her father to take early retirement, and by making him a small financial allowance, she ensured that he dropped quietly out of sight. Whenever she was asked (usually by workmates who did not like her) ‘are you related to that Umbridge who used to mop the floors here?’ she would smile her sweetest, laugh, and deny any connection whatsoever, claiming that her deceased father had been a distinguished member of the Wizengamot. Nasty things tended to happen to people who asked about Orford, or anything that Dolores did not like talking about, and people who wanted to remain on her good side pretended to believe her version of her ancestry.
This view stuck with her throughout her life and she wasn't exactly shy about it...
After a glass of sweet sherry, Dolores was always prone to spout very uncharitable views, and even those who were anti-Muggle found themselves shocked by some of Dolores’s suggestions, behind closed doors, of the treatment that the non-magical community deserved.
She ended up hating Harry because he represented things she hated, not the least of which being a well respected student who enjoyed his time at Hogwarts. Dolores had the opposite career there and resented that.
Dolores’s appointment as Inquisitor at Hogwarts gave full scope, for the first time in her life, for her prejudices and her cruelty. She had not enjoyed her time at school, where she had been overlooked for all positions of responsibility, and she relished the chance to return and wield power over those who had not (as she saw it) given her her due.
Her quest for control also helped shape her vile views and made her essentially no better than the Death Eaters
Dolores has what amounts to a phobia of beings that are not quite, or wholly, human. Her distaste for the half-giant Hagrid, and her terror of centaurs, reveal a terror of the unknown and the wild. She is an immensely controlling person, and all who challenge her authority and world-view must, in her opinion, be punished. She actively enjoys subjugating and humiliating others, and except in their declared allegiances, there is little to choose between her and Bellatrix Lestrange.
Out of universe Rowling based Dolores on a few bad experiences she had with teachers and coworkers that outwardly portrayed themselves one way, but were wholly different inwardly.
A love of all things saccharine often seems present where there is a lack of real warmth or charity.
So Dolores, who is one of the characters for whom I feel purest dislike, became an amalgam of traits taken from these, and a variety of sources. Her desire to control, to punish and to inflict pain, all in the name of law and order, are, I think, every bit as reprehensible as Lord Voldemort’s unvarnished espousal of evil.
All above quotes are sourced from the Dolores Umbridge, Original Writing by J.K. Rowling, Wizarding World.