Snape was in love with a half-blood.
So why in the world would he become a death eater, and join a cult which desired to rule over, and indeed often kill, half-bloods and muggles?
Was this ever explained? I have not read the books.
Sirius' recollections of Snape as a schoolboy confirms that he was obsessed with Dark Magic. This was the basis for his association with the Death Eaters.
"Snape’s always been fascinated by the Dark Arts, he was famous for it at school. Slimy, oily, greasy-haired kid, he was,” Sirius added, and Harry and Ron grinned at each other. “Snape knew more curses when he arrived at school than half the kids in seventh year, and he was part of a gang of Slytherins who nearly all turned out to be Death Eaters.”
(Goblet of Fire, Chapter 27, Padfoot Returns).
Snape's friends also had a large part to play. They took what was already a pretty twisted fascination with Dark Magic and exacerbated it. They were a bad influence on him, egging him further and further down the road to full Death Eater membership. Snape was pretty blind to how Dark the magic of some of his friends was.
"... thought we were supposed to be friends?” Snape was saying. “Best friends?”
“We are, Sev, but I don’t like some of the people you’re hanging round with! I’m sorry, but I detest Avery and Mulciber! Mulciber! What do you see in him, Sev, he’s creepy! D’you know what he tried to do to Mary Macdonald the other day?”
Lily had reached a pillar and leaned against it, looking up into the thin, sallow face.
“That was nothing,” said Snape. “It was a laugh, that’s all-”
“It was Dark Magic, and if you think that’s funny-”
(Deathly Hallows, Chapter 33, The Prince's Tale).
This move into darker and darker magic was also what alienated Lily from Snape. When he called her a Mudblood she essentially severed the friendship out of despair at the path Snape had taken. She also calls him out for being part of a gang of known Death Eater sympathisers.
“It’s too late. I’ve made excuses for you for years. None of my friends can understand why I even talk to you. You and your precious little Death Eater friends - you see, you don’t even deny it! You don’t even deny that’s what you’re all aiming to be! You can’t wait to join You-Know-Who, can you?”
He opened his mouth, but closed it without speaking.
“I can’t pretend anymore. You’ve chosen your way, I’ve chosen mine.”
(Deathly Hallows, Chapter 33, The Prince's Tale).
Essentially, Lily was one influence on him. Dark Magic and his Death Eater friends were another. He tried to keep these two parts of his life intact, essentially playing them off against one another. But they were irreconcilable. He picked the Death Eaters, to his eternal regret. JK Rowling has addressed this inner conflict in an interview.
Nithya: Lily detested Mulciber and Avery. If Snape really loved her, why didn't he sacrifice their company for her sake?
J.K. Rowling: Well, that is Snape’s tragedy. Given his time over again he would not have become a Death Eater, but like many insecure, vulnerable people (like Wormtail) he craved membership of something big and powerful, something impressive. He wanted Lily and he wanted Mulciber too. He never really understood Lily’s aversion; he was so blinded by his attraction to the dark side he thought she would find him impressive if he became a real Death Eater.
(Leaky Cauldron Web Chat).
Snape's reasons for wanting to join Slytherin:
Let's start from here as I think his joining Slytherin house is important to him eventually becoming a DE
From Snape's memory in OOtP we know that his father abused his mother, and possibly even him, and that he grew up in poverty. After he found out he was a wizard and would be going to Hogwarts, he was most likely very glad (just like Harry was) to be able to get away from his home. The fact that he and his mother were wizards while his father was a muggle gave him a way to differentiate himself from his father, and most likely even feel superior to him. Being an above average intelligence kid in a poor and abusive home, he desired wealth and power, both attributes generally present in a typical student from Slytherin. I think this is what made him want to join Slyhterin, even before he started going to Hogwarts (He tells Lily he hopes she goes to Slytherin as well)
Snape's reasons for wanting to join Voldemort:
Snape is sort of like a mini Voldemort, raised poor, muggle father not supporting his mother (and in Snape's case in fact, possibly physically abusing her), child prodigy, hated muggles and felt superiority over them. Whether he knew Voldemort was also a half blood at this point (he most likely knew after he rose through the DE ranks, for example, see here) is irrelevant. Voldemort's ideology already embodied many of the things he supported since he was a child. Add to the fact he grew up in Slytherin while in Hogwarts, where he is more likely to meet other students who would support Voldemort's ideals only helped solidify Snape's own beliefs (his first friend in Slytherin in Lucious Malfoy, see here).
I think it's only natural that Snape would eventually join the DE