R E L A T I O N S H I P SDON'T READ any of the spoilers below if you don't want it to be spoiled, specially those who have NOT read the manga or watched the anime, this is only for those people who don't really understand their relationships.....
The two actually have a hate-hate relationship, but they tolerate each other for the sake of Yuki Cross.
Zero Kiryu possesses complete distrust of Kaname Kuran's motives, even so far as to beg Kaien Cross not to trust him. At their first meeting, Zero attempts to stab him when he learns that Kaname is a Pureblood vampire. Zero later admits to himself there were a few possible reasons why he got so angry at Yuki's affection for Kaname. When Yuki expresses concern for Zero regarding the tattoo to Kaname, Kaname begins to get jealous of Zero for the first time.
The start of the story
Kaname reveals his awareness of Zero's inevitable awakening as a vampire and asks Kaien to move him to the Night Class. He also expresses respect for Zero's strength of will to resist the vampire instincts for four years. However, he does taunt Zero knowing his secret torture. Zero continues to be rude to Kaname and make him keep his distance from Yuki, his tolerance of Kaname has increased and he manages a reasonable level of civility.
After Zero's full awakening
Kaname expresses great disgust for Zero succumbing to his vampire nature and biting Yuki. Kaname gets so angry that Yuki is reminded of the first vampire she saw him kill. Kaname however quickly regains his composure and repeatedly reminds Zero that he owes Yuki for what she has done for him.
Kaname has already decided that Zero is his tool to protect Yuki. He advises Zero that despite his anger over his drinking Yuki's blood, he is allowing him to live and be Yuki's shield. When Shizuka Hio as Maria Kurenai endangers Zero, Kaname only expresses a little regret for the loss of his tool, believing Zero expendable.
Zero figures out that Kaname was the one who killed Shizuka after he gets confused over Kaname saving him from the council's vampires. He confronts Kaname to find out why the pureblood killed Shizuka and is surprised that Kaname wanted him to drink Shizuka's blood and even expresses envy that Zero is able to protect Yuki. Zero chooses not to reveal the truth about Kaname. Overall, Zero's civility has increased towards Kaname, though it always borders on rudeness.
Kaname's jealousy of Yuki's attachment to Zero causes him to be ruder to the boy and go as far as naming him a suspect in 25th Night when a human girl is being bitten. Though Zero denies it, Yuki is more offended by it than Zero is. Despite Kaname's accusation, he fully trusts Zero to protect Yuki to the best of his ability, reminding Zero of this on multiple occasions.
When Kaname convinces Zero to bite him, he expresses his most honest thoughts. Kaname tells Zero he hates him and causing Zero to shoot at the side of his head. Kaname slashes Zero's neck and he reveals how often he has thought about killing Zero with a single blow, but does not want to release Zero from his suffering. Zero only submits to taking Kaname's blood in order to stop hurting Yuki.
After Yuki becomes a vampire
Zero's anger toward Kaname is back stronger than ever. Compiled with all of his issues regarding Purebloods, Kaname is his foremost living enemy now. The two attack each other after Rido Kuran was killed, but he said that he wants to kill Yuki as well. Kaname gets mad because Zero pointed his gun at Yuki.
After the year
Zero spent the last year trying to track down Kaname, holding him responsible for the chaos that followed after destruction of the council, but Kaname proved to be elusive to Zero. When Kaien revealed the truth, Zero relented and allowed that Kaname is needed to keep the vampires in line. The two characters have had little contact since then.
Don't use this account anymore. Sorry