Alonso, the King of Naples, is not a particularly good guy, but not a particularly bad one, either. We know he was an enemy of Prospero, but the first we hear of the King is that he was easily swayed by Antonio's self-interested flattery. When we properly meet Alonso, we see he's kind of like any of the kids on My Super Sweet 16 – he's completely self-involved, easily moved to passion, sorrow, or tears, and even though he doesn't mean to be, sometimes he is a total jerk.
Alonso is easily moved one way or another, sometimes giving up his son for dead and other times searching for him doggedly. Gonzalo can sway him in one direction (towards good) when he speaks, but we know Antonio's wicked flattery also worked on the King before. That Alonso keeps Antonio and Sebastian, willing traitors, so close to him is evidence that he is at once trusting and naïve, in addition to being a horrible judge of character.
Unlike many of the other characters here, Alonso is quick to admit when he has done wrong – so long as he is called out on it first. When Ariel as a harpy reminds King Alonso what he's done to Prospero and Miranda, the King is genuinely sorrowful. Further, when Alonso sees Prospero, he's quick to return the man's dukedom. Yet we get the sense that Alonso doesn't think too much about his actions until he's called to account for them.
Because of his remorse and his willingness to embrace Miranda, his son, and Prospero, Alonso seems to be a not-all-that-bad kind of guy, just easily influenced by the wrong crowd. Most importantly, Alonso doesn't really trust his own senses. At the end of the play, he wonders at his son and can't really wrap his mind around the strange story they've all been part of. Ultimately, he's just another one of Shakespeare's misguided royals, not the brightest crayon in the box, easily persuaded, but not altogether bad.Timeline