Initially, the Ninja and Lloyd had a bitter relationship due to the fact that they prevented him from ruling Ninjago. When he redeemed, Kai realized his destiny was to protect him. Though Kai was jealous that Lloyd was the Green Ninja instead of him, that doesn't hinder their friendship.
In Season 4, when Lloyd's father was banished to the Cursed Realm to defeat Chen, Kai made a promise to look after Lloyd when Lloyd was sad about his father's sacrifice and demise.
In Season 5, after Lloyd was possessed by Morro, Kai was the most determined out of all the Ninja to get their friend back. When Morro left Lloyd's body and threatened him, Kai gave up the Realm Crystal and leapt into the water (despite his fear and inability to swim) to save Lloyd from drowning. He was relieved to have his friend back and comforted Lloyd when he was upset about dropping his guard by telling him that they would defeat any threat. Kai's words proved to be true as the Ninja were able to save the city of Stiix, defeat the Preeminent, and her army of ghosts.
In Season 7, he and the other Ninja come to the decision to make Lloyd their master after Wu is lost in time with the Time Twins. Kai most likely considers Lloyd as a close friend or even best friend following Season 5.