We know that Voldemort is a half-blood. He also hates half-bloods, we can see this from Harry Potter and the Half Blood Prince (Movie). When he takes command he starts killing half-bloods as well.
What is the reason to hate half-bloods even if he was a half-blood?
There were no targeted killings of half-bloods when the Dark Lord gained control. When the Dark Lord took power, he didn’t have half-bloods killed - he had Mudbloods, wizards born into Muggle families, locked up and sometimes killed. Those who could prove they had one close wizarding relative were spared.
“Nevertheless, unless you can prove that you have at least one close wizarding relative, you are now deemed to have obtained your magical power illegally and must suffer the punishment.”
- Harry Potter and the Deathly Hallows, Chapter 11 (The Bribe)
This would of course include all half-bloods, as all half-bloods have at least one parent who’s a wizard. Proving this further, one wizard in the hearings determining blood status claimed he was a half-blood as an attempt to get himself freed.
‘No, no, I’m half-blood, I’m half-blood, I tell you! My father was a wizard, he was, look him up, Arkie Alderton, he’s a well-known broomstick designer, look him up, I tell you – get your hands off me, get your hands off –”
- Harry Potter and the Deathly Hallows, Chapter 13 (The Muggle-born Registration Commission)
If half-bloods were hated under the Dark Lord’s rule, this wouldn’t help, and would be useless saying. The Dark Lord didn’t have half-bloods killed for being half-bloods. He surely would kill a specific half-blood if they were in his way - he was willing to kill anyone if they were sufficiently in his way. However, it’s clear that they weren’t actively tracked down by his Ministry to be removed from wizarding society by any means. In the official Ministry position, wizards with no proven wizarding ancestry were said to have stolen their magic.
“Recent research undertaken by the Department of Mysteries reveals that magic can only be passed from person to person when wizards reproduce. Where no proven wizarding ancestry exists, therefore, the so-called Muggle-born is likely to have obtained magical power by theft or force.”
- Harry Potter and the Deathly Hallows, Chapter 11 (The Bribe)
Half-bloods do have wizarding ancestry, though it’s not pure wizarding ancestry. So, this wouldn’t apply to them. Half-bloods were never targeted. There were Ministry rewards for capturing Mudbloods and blood traitors (pure-bloods who didn’t value their blood status) but none mentioned for half-bloods.
“Snatchers,’ said Ron. ‘They’re everywhere, gangs trying to earn gold by rounding up Muggle-borns and blood traitors, there’s a reward from the Ministry for everyone captured. I was on my own and I look like I might be school age, they got really excited, thought I was a Muggle-born in hiding. I had to talk fast to get out of being dragged to the Ministry.”
- Harry Potter and the Deathly Hallows, Chapter 19 (The Silver Doe)
Hermione says it’s only Mudbloods the Death Eaters hate, and there aren’t enough pure-bloods for them all to be pure-bloods. So it’s likely that the Dark Lord allows half-bloods into the Death Eaters as a general rule, not an exception - which he wouldn’t if he hated them like he hated Mudbloods.
“The Death Eaters can’t all be pure-blood, there aren’t enough pure-blood wizards left,’ said Hermione stubbornly. ‘I expect most of them are half-bloods pretending to be pure. It’s only Muggle-borns they hate, they’d be quite happy to let you and Ron join up.”
- Harry Potter and the Half-Blood Prince, Chapter 12 (Silver and Opals)
Also, it’s further implied that half-bloods are allowed to be Death Eaters, as J.K. Rowling says Snape being a Death Eater is a hint to his ancestry, because Mudbloods are rarely allowed to be Death Eaters.
Apart from Harry, Snape is my favourite character because he is so complex and I just love him. Can he see the Thestrals, and if so, why? Also, is he a pure blood wizard?
Snape’s ancestry is hinted at. He was a Death Eater, so clearly he is no Muggle born, because Muggle borns are not allowed to be Death Eaters, except in rare circumstances. You have some information about his ancestry there.
- Edinburgh Book Festival (August 15, 2004)
This implies that his ancestry isn’t unusual among the Death Eaters, further evidence that the Dark Lord only hates Mudbloods, as he wouldn’t allow half-bloods to join the Death Eaters if he hated them.
He doesn't?
There is no evidence that Voldemort was specifically targeting half-bloods. While he certainly considered pure-bloods to be of a higher standard, he didn't even hide his half-bloodedness from the Death Eaters - he is telling the story of his parents to Harry as the Death Eaters arrive (so they can hear him as he's talking to Harry).
Voldemort laughed again. Up and down he paced, looking all around him as he walked, and the snake continued to circle in the grass.
"You see that house upon the hillside, Potter? My father lived there. My mother, a witch who lived here in this village, fell in love with him. But he abandoned her when she told him what she was.... He didn't like magic, my father...
"He left her and returned to his Muggle parents before I was even born, Potter, and she died giving birth to me, leaving me to be raised in a Muggle village... but I vowed to find him... I revenged myself upon him, that fool who gave me his name... Tom Riddle...."Still he paced, his red eyes darting from grave to grave.
"Listen to me, reliving family history...." he said quietly, "why, I am growing quite sentimental... But look, Harry! My true family returns..."
The air was suddenly full of the swishing of cloaks.
[...]
"I knew that to achieve this - it is an old piece of Dark Magic, the potion that revived me tonight - I would need three powerful ingredients. Well, one of them was already at hand, was it not, Wormtail? Flesh given by a servant...
"My father's bone, naturally, meant that we would have to come here, where he was buried. But the blood of a foe...["]
Harry Potter and the Goblet of Fire, chapter 33: "The Death Eaters"
And he had half-bloods in his innermost circle, such as Severus Snape.
Half-bloods were allowed to keep their wands in Harry Potter and the Deathly Hallows, while Muggle-borns were not. The pureblood Wizarding community was so small, that if he had killed all half-bloods, the Wizarding community would soon die out, because there were so few people.
So, to answer the question "Why does Voldemort hate half-bloods?" - he doesn't; he just holds them to be lower than purebloods (but still higher than Muggle-borns).
In fact, he apparently doesn't want to kill even all Muggle-borns - seeing how many times he tells Lily Potter to "step aside". He says in Deathly Hallows that he doesn't want to spill magical blood:
"If you continue to resist me, you will all die, one by one. I do not wish this to happen. Every drop of magical blood spilled is a loss and a waste."
Harry Potter and the Deathly Hallows, chapter 33: "The Prince's Tale"
And earlier in Deathly Hallows, we see that being a half-blood under Death Eater rule is better than a Muggleborn:
And then, abruptly and shockingly amid the frozen silence, one of the dungeon doors on the left of the corridor was flung open and screams echoed out of it. "No, no, I'm half-blood, I'm half-blood, I tell you! My father was a wizard, he was, look him up, Arkie, Alderton, he's a well-known broomstick designer, look him up I tell you — get your hands off me, get your hands off —" Harry Potter and the Deathly Hallows, chapter 13: "The Muggle-born Registration Commission"
An alternate take on Mithrandir's answer: He doesn't.
Voldemort wants power. Well, that and eternal life.
Leading a group of Purebloods with large bank vaults (and large libraries of magical tomes!) on an Blood Purity crusade seems to give him that, and gives him an excuse to vent against the Muggles he blames for abandoning him and his mother.
(He shows resentment towards his Muggle father for leaving his mother and not trying to find him, and towards the Muggle matron of the orphanage. Delving into speculation, he may blame the Muggle world in general for "abandoning" orphans in squalid orphanages, and however his father and grandparents may have reacted when he visited Riddle Manor and killed them. After all, if they'd greeted him with "Well, you're handsome, suave, intelligent, in possession of useful and esoteric skills - and we currently lack an heir. How would you like to become future Lord of the Manor, and eventually join the Government as a influential Peer?" then he'd most likely have said "Awesome - how about I run for Prime Minister and pull that pesky Ministry of Magic under my control?")
However - look at the way he treats the Death Eaters throughout the books with distain, leaving them cowering in fear. Look at the way he treated the Gaunts - murdered his Grandfather, and framed his Uncle. He leads the Purebloods on a self-destructive campaign that turns society against itself, and actively targets Pureblood families such as the Bones or Weasley families. Most of the actual discrimination against Muggleborns or Halfbloods is carried out by his followers or their allies (*hem-hem*Umbridge*hem-hem*), without much input from Voldemort himself.
All in all, his actions seem to indicate that the "Pureblood cause" is little more than a convenient rallying banner to gather support under for his own aggrandment, rather than a conviction that he himself holds.
He is blood purist.
I am not completely sure about Mithrandir's opinion, that Voldemort didn't hate half-bloods, but it seems likely that he did hate them less than Muggle-borns.
I didn't evidence of his own opinion but at least his closest death eater Bellatrix made her opinion very clear (we can assume, that she got most of her opinions from Voldemort):
“Shut your mouth! You dare speak his name with your unworthy lips, you dare besmirch it with your half-blood's tongue, you dare”
― Bellatrix Lestrange, Harry Potter and the Order of the Phoenix
Voldemort himself was a blood purist, he dreamed about having a wizard world with only pure-bloods, therefore any disruption of the blood purity was a matter of hate for him.
Also, Ron seems to think that half-bloods aren't exactly liked by the purist community:
“Dirty blood, see. Common blood. It’s ridiculous. Most wizards these days are half-blood anyway. If we hadn’t married Muggles we’d’ve died out.”
― Ron Weasley, Harry Potter and the Chamber of Secrets
Also, I would speculate, but I think, that the fact he himself is half-blood only increased the hate inside him (and probably also towards himself). He wasn't exactly balanced person…
