I should have saved the explanation I typed out last night, when someone asked the same question, at the exact same time of night.
Lelouch was inadvertently caught up in the mass slaughter of Elevens, and the fact that his brother condoned that kind of behavior combined with the fact that Lelouch had spent many of his formative years fermenting a virulent hatred for Britannia and his family that cast him out made him think all of five seconds about killing Clovis. To Lelouch, Clovis represented all that he hated, not only being a cruel noble who preyed on the weak but also someone who retained what he himself was robbed of. Clovis loved Lelouch, but relating any of his feelings to Lelouch at that point would have sounded like empty flattery and pathetic begging for his life to Lelouch.
He also did it to test himself. Lelouch threw up when he got back to school, proving he didn't like what he did. But he did it to prepare himself for when he would have to face his other, stronger siblings so that he would be able to confront them without fond memories of them clouding his judgment.