Kinzou's (earlier GIII) relationship started off antagonistic due to ordering the attack on Team Baskerville causing Kinji to seek vengeance against him, which later was revealed as an attempt to force Kinji to enter King HSS mode. During the final battle between them, Kinzou revealed that he is in fact Kinji's younger brother causing Kinji to save Kinzou despite everything he had done to him.
Following their fight and Kinji's transfer to a normal school, Kinzou attitude while still being a tsundere had changed with him looking up to Kinji and envying what he has tried to obtain by wanting to live the peaceful, boring life. Also, despite Kinzou prideful nature on the outside, he is a caring and supportive person to Kinji by accepting Kinji decision to transfer out of Butei High, following him while camouflaged to Kinji's new school to look after him, and even teaming up with Aria to save him, Moe, and Kikuyo from the Kagatake Family Syndicate.
It's unknown how much Kinzou and Kinichi interact with each other, except they have some sort of understanding as shown in Volume 11.
Kaname and Kinzou are siblings, and although he doesn't show it very often, he does care about Kaname's well being and tried to not lead her more down the path of destruction then he already has. As shown when he broke ties with her and started calling her "Kaname" instead of "Fourth".
Kinzou wanted to use Aria's power to go back in time to save professor Sarah.
Magane is Kinzou biological grandfather, who he seems to respect as a warrior and constantly tries to get him to teach him a 'Secret Technique'. While Kinzou attitude is still the same, he calls Magane 'Gramps' and defers quite readily to the older man out of both fear and respect.