Here's my interpretation of the story. The prince was a minor prince with no power and no desire to become emperor. But the most powerful prince still wanted to eliminate his rivals, so he had his tea poisoned. But that day the princess drank it first and died instead. He blames himself for not fighting for the crown and being so powerless to protect the people he loves. Kiel and him also grow apart because of this incident, partly because the prince blames him, and partly because tragedy often does that to friendships.
When he brings his sister to the Emperor and asks him to punish the person responsible, the Emperor doesn't care and wouldn't punish the (at that time) Crown Prince for a mere princess and powerless prince. So he decides to pursue a bloody path of revenge and become Crown Prince to gain the power to punish his brother. He probably killed his brother, hence the name "Blood Prince." Due to this incident, the Prince absolutely despises callous royalty and believes that strict nonviolence is just cowardice that puts good people in danger. He likely feels no guilt in murdering the royal family of Marie's kingdom, who he called cruel previously. With Marie though, he's reminded of his sister, and would rather not have to kill an innocent royal with a benevolent reputation in pursuit of power.
Because Kiel and the Imperial Knights swear fealty to the Emperor, Kiel feels responsible to his family and position to oppose the Prince, even though personally he would rather stand at his side. To the Prince though, he doesn't want to kill his friend, and is angry that Kiel is stubbornly being a slave to the comatose Emperor who let his sister die. Hence the current dilemma. An all-round tragic circumstance, and I can understand why both feel this way.