Let's start by making one thing clear: Shakespeare's King John is not a Renaissance prequel to the Flipper movies, and Louis the Dauphin is not a fun-loving sea creature.
The name "Dolphin," more commonly used with the French spelling of "Dauphin," is just a title that is given to the Prince of France, specifically the guy who's next in line to inherit the throne. (It's just like how the guy in England next in line to inherit the throne is always called the Prince of Wales.)
Are we on the same page? Good.
As for the specific personality of Louis the Dauphin in King John, it's not very complicated. Basically, Louis is your typical belligerent hotshot, always spoiling for a fight. Like most belligerent hotshots, Louis is also deeply narcissistic, as you can see in his ridiculous attempts at charming Blanche—by praising his own reflection in her eyes.
Yeah, fellas, we don't recommend taking any romance tips from any guy who calls himself a dolphin.
Anyway, in the end, Louis's total disregard for others ends up leading to his undoing. When Salisbury, Pembroke, and Bigot (his English allies in the war against King John) learn that Louis is planning to have them killed after the war, they understandably abandon him. This totally messes up Louis's plans of destroying John and taking over England, and it forces him to head back to France.
Louis doesn't seem too put out by this, though. After all, there will always be more battles to fight, and more countries to invade. At least that's how he sees it, we guess.