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The story itself never goes into detail about Seishirou's relationship with the twins, but the fact that the twins paid an apparently high price to travel the dimensions to avoid him suggests that it did not go well. What is known is this: Seishirou met the twins at some point and Subaru gave him his blood. At some point later, it became necessary for Subaru and Kamui to run away from him, so they went to the Space Time Witch and made a deal of an unknown, but definitely costly value in order to travel between dimensions. Seishirou also went to Yuuko's shop and traded his eye for the ability to travel through dimensions as well. He will find the Vampire Twins, though his specific interest seems to be in Subaru, even if he has to destroy other universes to do so.
Much of the rest of the story remains vague or unknown. For example, why did Subaru give him his blood? Why did Kamui and Subaru need to flee him? What did he do (or not do) that made Kamui so furious at him? Does he plan to harm the Vampire Twins? How does he feel about Subaru exactly? How does Subaru feel about him? To what degree does jealousy motivate Kamui? The questions could go on and on.
What is most significant for the story is that Seishirou will go to any lengths to find them. At one point, when Syaoran was still a kid, Seishirou trained him (or the Syaoran clone anyway) in his fighting style in exchange for translating a book on vampires that Syaoran had in his possession. Since both Seishirou and Syaoran are both missing their right eyes, the fighting style they both needed was very distinct. Naturally, learning from Seishirou changed his life and allowed him to be more of aid to Sakura, particularly on their trek. Still, despite their history as master and pupil, Seishirou was willing to hurt Syaoran if he got in his way. Finding the vampires is paramount.
Seishirou is perhaps the most important person in Kamui's life, with the notable exception of Subaru. While Kamui may fear for his own existence, he likely also fears for the bound he shares with his twin. Interestingly, when Hokuto faced the same problem (Seishirou was planning on harming her brother), she faced him in order to place a special curse on him that would keep him from killing her twin. In this story, Subaru's twin pulls Subaru into a potentially deadly game of hide-and-seek/tag through the dimensions to keep him safe from Seishirou's influence. Whatever plans Seishirou has for the pair of twins,
While Kamui very clearly hates Seishirou, the Hunter says very little about him. When he talks about Kamui, it is always as part of the set, rather than an individual. The only comment about Kamui made by Seishirou comes from second hand from Fuuma, who says, "You're just as big brother described you. You are strong, aren't you, Kamui?" (vol 17) What exactly Seishirou said about Kamui in terms of his description remains unknown.