We do not see much of Kyougo as character in X. For the most part, he appears to be more of a plot device, in that very seldom do we see much about his motivations... although we can certainly infer quite a bit.
Going by his first appearances and interaction with Fuuma and Kotori, we are painted a picture of a kind, caring father. He is clearly proud of both of his children. There is also a certain sadness to him, which we assume at first is due to his wife's death years ago. It is also clear that Fuuma and Kotori love their father. This is a family that struggled to come together and overcome the mother's death, and that they did so together: their father's love and support in particular made Kotori and Fuuma grow into "the pride of the town". However, it is equaly clear that due to Kotori's fragile health (both mental and physical), it was Fuuma that bore the greater part of the burden. It was undoubtably Fuuma that Kyougo relied on.
This fact is further stressed after Kyougo's death: while talking to Kotori, Fuuma tells her that if it hadn't been for her, he might have done "something disgraceful enough to drive father wild". While this also sheds light on what Kotori was to Fuuma, it is equally indicative of how important it was to him to meet Kyougo's expectations. Kyougo was... what a japanese man should be, and I'm sure Fuuma wanted to be that as well. His father was a role model for him, and no doubt at least part of the reason why Fuuma is who he is at the beginning of the series: serious to the point of being stern, protective of those that needed him, and very mindful of his duty as son, brother and student.
But Kyougo was much more than just the father Fuuma knew. There is his relationship with Saya, being willing to live with the knowledge that she loved someone else, yet accepting that because of fate, duty and love. And then of course, there is that duty, a duty that neither Fuuma nor Kotori knew about until it was too late. Kyougo, as priest of Tokagushi shrine, was the guardian of the Shinken. He also knew, to some extent, what Fuuma's fate would be; yet he kept the truth from Fuuma until the last possible instant. In that single fact, we can see the dichotomy that Kyougo was: one part loving father, who wanted nothing more than to keep his children safe and happy; and one part the dutyful servant of destiny, carrying out his duty to the best of his ability, willing to sacrifice his life for it, and knowing that ultimately, he must pass the burden along to his son.
Now, to all outward appearances Fuuma is following in his father's footsteps: in becoming the kamui of the Angels, he is also dutifully doing what is required and walking the path set for him. However, it's my belief that even as he apparently does so, he is also staging a revolution.
Saya is one of the most overlooked characters in the entire series, in part, because her role is minimal in the sense that she is already dead when X starts and we only ever see her through a few flashbacks. As far as the impact that she had on certain key characters and on the story itself... in a very real sense, there would have been no story if not for Saya. Or at the very least, things would have been very different. The impact that her choices and her ultimate fate had on Fuuma, specifically, was huge but also very subtle.
Even though it has never been stated outright, to Fuuma, Saya is a symbol of the innocence and happiness that he has lost.Yes, in the page before he was talking about how he was happy as long as Kotori and Kamui were happy, but those memories are linked to the memory of Saya and Tooru watching over the three of them together. And of course, Saya's death changed everything, so it's logical that in his mind, happiness might be at least subconsciously linked to the time when Saya was alive.
If Fuuma wanted to be a "good man", the kind of man his father was, nowhere is that desire more evident than in his relationship with his younger sister Kotori.
One of the roles that Fuuma took on as his own, even before his mother's death, was the role of protector. In many ways, he is still a protector now (what a lot of people don't understand about him is that as the twin star, the Kamui of the Dragons of Earth and as so-called "wish genie", Fuuma continues to fullfill the role of protector.) Before he was anything else, Fuuma was his little sister's champion. Oh, he became Kamui's as well, but their childhood promise makes Fuuma's priorities crystal clear: he would protect Kamui as long as Kamui never made Kotori cry. The point of the promise was not to safeguard Kamui, even though he was important to Fuuma too; the point was always Kotori. He was her older brother, and he was supposed to watch out for her. As simple as that.
This becomes even more crucial after Saya's death. It was their mother's violent, gruesome, and seemingly senseless death that unhinged Kotori, it damaged her irrevocably, and it was then that Fuuma really shouldered the burden. There are several accounts of what may have happened after Saya's death. Two of those (Kamui or Fuuma seeing his counterpart pick up Saya's head and lick her blood off his hand) are meant to be taken as metaphors, as dream images. I believe that what really happened is what is shown in volume two: Fuuma and Kotori unwittingly walk into a room splattered with Saya's dismembered remains. Kotori can't handle the sight, she completely loses it; and Fuuma does what he has to to do, the only thing he can do: he holds it together, he hugs Kotori and tries to shield her from the truth, he insists that that bloody mess is not their mother. Over the next seven years, Fuuma continues to do the same thing. He is his sister's silent champion, he's there to protect her both from the world and from that memory and the darkness that it brings.
So if Fuuma's one purpose since he was small was to protect Kotori, how is it that he ended up killing her, and in the way that he did?
This has never been truly explained. On the one hand, we know that Kanoe showed Fuuma dream images of Kamui killing Kotori in the hopes that it would influence Fuuma and bring him to her side. It would seem that that, coupled with the sheer power, awareness or whatever else that was released with Fuuma's awakening as the other Kamui pushed Fuuma to kill the one he wanted to protect.
But consider one thing: it was Kotori's wish to die, if it would mean saving the earth. Although at the time that he killed Kotori we had seen no other evidence of it, Fuuma's fundamental power is that he sees people's wishes, and where he can, he grants them. It is, therefore, quite likely that he killed Kotori in response to her wish. It is also entirely possible that in a moment of suddenly heightened awareness he realized that it was destined that he or Kamui had to kill her, and that he did it so Kamui would not. Why? Perhaps it was to spare Kamui from having to do it. Or perhaps the truth lies in a conversation he has much later with Kakyou: "if there was someone that was ultimately important to me, I wouldn't allow them to be killed by anyone else". Everyone always assumes those words refer to Kamui. They could, however, mean other people. They could mean Kotori.