Since the young King is so keen on his no-girls-allowed secret society ("the Academe"), let's analyze him in terms of a boy's club. The King is the Alpha. He's a natural leader; the other guys do what he says. Motivated mostly by a desire for fame (or maybe out of a some kind of fear), the King proposes three years of uninterrupted study. His cronies Dumain and Longaville quickly agree, and, after some grousing, so does his right-hand man, Berowne.
The King is a perfectly respectable guy, polite and responsible. He lives in the beautiful, idyllic kingdom of Navarre where he can choose to spend his days hanging out with his buddies and keeping his nose in a book (he's got a library that would make Amazon.com jealous). There's nothing too worrisome troubling Navarre or its inhabitants. In essence, the King is a lucky ruler with a lot of free time on his hands.
When he first meets the Princess of Navarre, we've got to believe he does a double take. Can't you just see it? This boy's club president is suddenly disarmed, and perhaps realizes instantly that he's not as mentally tough or committed as he thought he was. He apologizes profusely to the Princess for denying her admittance to his house, and treats her with respect in their discussions of Aquitaine. He's conventional, too. Take a look at his love letter to the Princess. Rather than try to dazzle her with his quick wit, or sophisticated style, he just repeats, over and over in different ways, that he's been crying a lot.
As befitting a leading man, the King changes over the course of the play. He learns. Just take a look at his first speech. He proposes this scheme of foregoing women, food, and sleep, and equates these vows with heroism. He and his friends are "brave conquerors… / That war against your own affections / And the huge army of the world's desires" (1.1.8; 9-10). For the King—before he meets the Princess—honor and desire are mutually exclusive. He must divorce himself from instinct because it gets in the way of what he wants: glory and fame. Navarre must be "the wonder of the world" (1.1.12). Why? So the King can live forever:
When, spite of cormorant devouring time,
Th' endeavor of this present breath may buy
That honor which shall bate his scythe's keen edge
And make us heirs of all eternity. (1.1.4-7)
He's going to cheat death by studying his way to fame. (Oh yeah; that'll totally work.) If you take this idea seriously, the King's struggle not to fall in love suddenly seems less silly. He's afraid that loving will make him die forgotten. At the end of the play, the Princess—newly acquainted with the death of her father—seems to think he should confront that possibility. She asks him to accept obscurity: a year in a "forlorn and naked hermitage" (5.2.369). No longer centered on himself, the King agrees to the Princess's proposal, and even imagines himself as a part of her:
KING. If this, or more than this, I would deny, To flatter up these powers of mine with rest, The sudden hand of death close up mine eye! Hence hermit then, my heart is in thy breast. (5.2.872)
But we're still left with lots of questions at the close of the Love's Labour's Lost: will the King and the Princess eventually reunite? Does the King really love the Princess, or is just happy to feel the joys of being in love? And lastly, how well do we really know this character?