He's a complex portrayal of a man tortured by his own genius, one who seems to be ashamed of his own mind and yet covers it up with what seems like unyielding certainty and pride. But as we can clearly see he is led by strong emotions most of the time and is capable of impulsive acts, especially when he is in a moment of passion or anger (ex.during his conversation with Alyosha in the tavern). It almost seems as if he forcefully tries to put his brain above his heart, but is unable to, which creates an odd balance between the two. His beliefs exist solely because of how deeply he cares — his anger at God is caused by his rage at human, especially children's, suffering. The logic behind it is quite simple, if a God who is portrayed by the church as pure good allows cruel things to happen to innocent souls, He either does not exist or isn't worth praying to. Although this vision of God might seem egoistical — for humans to possess free will and yet blame God for all the cruelty in the world — it is a reasonable conclusion, one that someone desperate for an answer and of his intellect would come to. He believes that when Christ refused to offer humanity "happiness and safety" and therefore making the whole humanity "bow down to him", the church, Inquisitor, stepped in to offer said things, since "humanity will always feel the need to bow down to someone" who "provides the bread" (heavily paraphrasing). His beliefs are obviously not praised in the book, seen as more or less of some sort of a 'Western trend' that has possessed Russian intellectuals (which is influenced by Dostoevsky's personal attitude towards faith) but they are not villianized either, since he is given a 'human side' unlike, for example, Smerdyakov (due to the fact that Dostoevsky most likely spread his own personality pieces between all the characters, seemingly excluding Smerdyakov). Perhaps it is because he is open to an concept of God, not a purely good one, but still a God. The frist thought of his we encounter, "if there is no God everything is permitted", doesn't seem to be a ideal of his, since we clearly see that not only he, again, doesn't deny that there is a God in his further words and works, but also doesn't possess the lack of morality this line would imply, since he doesn't excuse Smerdyakov's actions with it (although it could come from his personal complicated 'hatred' for the man) nor he excuses the things people do to children, when he brings the topic up to Alyosha.
When it comes to his feelings towards other characters, the only ones that seem clear to me are his feelings towards Katya and Alyosha. He clearly deeply loves both and desires to be loved back by them, although in different ways, one being love in the romantic sense and the other being in the platonic sense. He desires to be understood by each of them, although he only directly expresses it to Alyosha in the book. When it comes to his relationship with his father we can only assume he had an internalized hatred for him, which is both shown in his approach to him and his certainty in being trailed alongside Smerdyakov after he confessed his crime to him — obviously, it was due to Smerdyakov's manipulation, but would a man of his intellect truly believe he wanted his father dead, if he didn't? Sickness might've taken a toll on him, but I'd say the desire for the death of their father is present in every Karamazov sonz excluding Alyosha. His relationship with Mitya doesn't seem as deep as his bond with Alyosha, perhaps due to growing up apart or quite large difference in personality and I'd risk saying that the escape plan he has prepared for Mitya comes from his underlying need to take responsibility over the situation, not brotherly love. His relationship with Smerdyakov is an even more complicated one — on one hand, we cannot deny the mutual hatred between them, on the other hand we cannot say it is only what Ivan feels towards him. He seems to be drawn towards Smerdyakov and despite his best efforts he always ends up being less harsh than he would like. Due to his general attitude, it's almost impossible to tell if it's a strange affection, an underlying sense of kinship or he's simply pleased that the latter admires him and his ideals — all of those possible aspects may be tangled with each other and looping back to his pride.
