The hat wanted to put Harry in Slytherin too, as has been mentioned above, but what hasn't been mentioned is that Harry is a half-blood as well, what with his mother being muggle born.
That's probably why Voldemort believed him to be the wizard the prophecy was about.
If Harry is a half-blood and he almost got put in Slytherin doesn't that prove that you don't have to be pure blood to get into Slytherin? It just helps you get there.