We don't learn too much about Kain at first. After all, he does disappear right after you fight the very first boss in the game. He's definitely something of an enigma. It's apparent straight away that he is a proud man, and he exudes confidence and control. Obviously respected throughout Baron, Kain holds his head high. However, he can come across as cold and aloof, and has a definite sarcastic and snarky streak running through him. But Kain isn't unfeeling and it's not that he doesn't care about others. Clearly he does - he put himself on the line to stand up for Cecil when the king demotes the Dark Knight for his supposed disloyalty. He may have dashed into the throne room on impulse, but Kain could not just stand by and say nothing while his best friend was being treated so poorly.
Cecil may be Kain's best friend, but the Dragoon keeps even him at arm's length. Kain plays his cards close to his chest, keeping his feelings to himself and only rarely letting his guard down. Not even Cecil knew of the reason why Kain became a Dragoon until Kain tells him in a moment of reminiscing after they leave the throne room. And these two grew up together! (Although is is probably just exposition for the benefit of the player.) Kain is probably something of a loner; the beginning of Final Fantasy IV Advance shows him training alone in the mountains. He is actually quite passionate, but just not in a very overt manner. Kain is a dragoon, and being a master dragoon is incredibly important to him. Despite pressures to become a dark knight like Cecil, Kain has dedicated himself fully to following in his father's footsteps. To Kain, surpassing his father as a dragoon is his way of honouring him, and he wants to make him proud. Kain's honour, and the honour of the dragoons, are also hugely important to him.
Kain's feelings for Rosa are probably one of the main reasons Kain doesn't get too close to people. Kain is in love with her, and is jealous of her relationship with Cecil. Both Kain and Rosa were born to noble families, and it would have been assumed that they would end up together. Cecil, while being raised by the King, was of unknown parentage. Kain's pride must have taken a huge dent when Rosa picked Cecil over him. Whether Rosa knew of his feelings or not is debatable, not even Cecil knew, so it's possible Kain was just so good at concealment she had no idea. But unrequited love will eat away at you, especially when your closest friend has what you desperately desire.
But despite all this, Kain strives to be a good and loyal friend and keeps his emotions under check. Or tries, at least. When he and Cecil realise they were sent to Mist to destroy the town and the summoners living in it, they both know they can no longer serve the King. Kain agrees to help Cecil to try and stop Baron, and the latter is surprised when Kain shortly tells him that he isn't doing it for him.
Kain isn't evil, nor is he as admirable as Cecil. He is fundamentally flawed, carrying around a ball of resentment and bitterness inside that only grows as time goes on, and that nearly ends up costing him everything he holds dear. Worst of all, he betrays his friends and disgraces the dragoons. Kain's split personality brings to light all those emotions and feelings that he had tried to keep hidden from everyone. Everyone has darkness inside them, but Kain believes he is weak because these thoughts allowed him to be manipulated, so he sets about fixing that. By the end of the game, while Cecil and Rosa have long forgiven Kain for what he has done, Kain can't forgive himself. Conspicuously absent from their wedding, Kain pours his passion and drive into become a master dragoon, to face the trial that Cecil once did on Mount Ordeals and redeem himself.