Final Fantasy IV provides a happy ending for almost everyone. Most people seem to inherit a kingdom, wedding bells are playing, all those characters you thought were dead are in fact alive. The glaring exception is Kain. Conspicuously absent from the wedding of Cecil and Rosa, he has instead gone off to climb the mountain of his angst.
Our last glimpse of Kain is of him standing at a cliff on Mt. Ordeals. Unlike the others, who attended Cecil's and Rosa's wedding, Kain is here, giving a lovely soliloquy.
"Cecil, Rosa...forgive me once more. I cannot attend -- my actions have shamed the ranks of Dragon Knights everywhere. Mt. Ordeals tested you, Cecil. Now I must test myself. If I surpass my father as a Dragon Knight, his spirit might forgive me..."
This speech makes it clear that Kain can't forgive himself for all the happened during the events of the game. His loyalty to Cecil wavered several times, his pride as a dragoon was shattered, and he feels that his father must be frowning upon him. He doesn't care that it wasn't his fault that he was controlled -- he seems to see that as a flaw as well. He feels the need to test himself -- to become a pure, great dragoon -- in order to atone for his sins.
Kain's fate is undetermined- after all, Cecil was the only one able to complete the trials to become a paladin, and that might've had something to do with his space alien nature. Could Kain complete the test and become a weird dragoon-paladin thing? I don't know why he would want to, imagine how silly that outfit would look. He seems happy being a dragoon- indeed, the Dragon Knights are all he talks about. I don't think he's looking for the literal class-change dead father's spirit sanctioned forgiveness. But if it's not that, then what? Does gaining XP cleanse your soul?
Simply put, Kain needs some time alone to think. Battle is the one thing he knows. If he spends enough time at that, perhaps he can achieve forgiveness. Not from Rosa and Cecil- they forgave him a long time ago. He needs to forgive himself.