It never bothered him that he was a 'half-blood'. Why should it? He had no control over who his parents were.
But then he found the prince's potions book. It still didn't make much of an impact on him. But he couldn't deny that he empathized with the half blood prince. That was until he found out the prince was Snape.
It occurred to him that Voldemort was also half-blood. This bothered him more than the prince. He didn't want to have anything in common with that monster, but it seemed that all three men were more alike than different.