Initially, Kristoff viewed Anna as a nuisance and, presumably because of his dislike of people and lack of social skills, wanted nothing to do with her. After she showed him kindness by buying the mountain gear and carrots he was unable to afford, the mountain man was willing to return the favor by assisting the princess in ending winter; though this was implied to mostly be for the sake of his ice business. However, as the adventure continued, resulting in the loss of Kristoff's sled and supplies, Anna offered to repay him, but selflessly added an offer to relieve him from the journey, to avoid further turmoil. Even so, because of his ultimately soft heart, Kristoff decided to continue on, alongside the princess, though he tried to cover his softness by claiming to assist the former for the sake of a new sled.
Eventually, Kristoff's soft heart overtook his initially grumpy and selfish nature, showing genuine concern and care for the princess throughout the course of the film, and even found himself impressed by her fearless, and adventurous ways. These feelings would soon become romantic, though because he knew Anna was already engaged to Hans, Kristoff made an attempt to suppress these feelings.
Through Olaf, Kristoff's love for Anna was made apparent, and during the climax of the film, after originally leaving the kingdom after their adventures came to a close, Anna realized Kristoff willingly returned for her sake, proving Olaf's words to be true. By the end of the film, the two began a relationship; sealed with a kiss during the epilogue.
In Frozen Fever, Kristoff presents Anna her birthday cake as he proclaims his love for her, becoming a bit embarrassed at what he has just said while Anna returns that sentiment with an elated, heartfelt grin.
Allies and Friends
Sven is Kristoff's pet and best friend. They have known each other since childhood and are inseparable.
Kristoff has a habit of speaking for Sven in a goofy voice (much like many real pet owners do with their pets), but Sven doesn't seem to mind. Kristoff loves Sven a lot, and always make sure he has enough food to eat (carrots), even though they share their food with each other most of the time. Kristoff can perfectly understand what Sven is saying, which is why he does voice-overs for Sven and Sven often acts as his conscience, being the one to convince Kristoff to continue accompanying Anna on her journey so she wouldn't freeze to death, and Kristoff could get a new sled from her. Sven was also smart enough to realize Kristoff's feelings for Anna and tried to convince Kristoff to realize his feelings for her. When Sven and Kristoff are rushing to get to Anna, Sven accidentally falls into the water. Despite needing to get to Anna immediately, he looks back and waits a few moments to make sure Sven was alright before he rushes back to save Anna, indicating how much he cares for Sven.
Kristoff seems protective of his friend, such as the wolf ambush scene, where he's annoyed at Anna giving Sven the command to jump over a ravine, and gives the order himself after cutting Sven's harness. He also tells Olaf not to talk to his friend demeaningly, even showing hints of jealously towards others who make an attempt to befriend him.
Though Sven can at times be annoyed by Kristoff's selfishness, and Kristoff hates it when Sven acts as his conscience, Kristoff and Sven are very close and treat each other more like equals and siblings rather than a traditional master/pet relationship, with both caring for the other's happiness.
Kristoff and Olaf have somewhat of a good friendship and are friendly towards each other. At first, Kristoff calls him creepy, and is confused by the fact that Olaf (being a living, breathing frozen substance) would dream about summer and being in heat itself. He wants to break the ice and tell the happy snowman that his dream might not be all what he thinks it's cracked up to be, but Anna stops him by smacking him in the arm and scolding him. Kristoff still believes that somebody has to tell Olaf the truth about his dream eventually, but doesn't try to proceed to do it himself. However, he seemed to hold no animosity or dislike towards the snowman. He also warned Olaf to be careful and not be seen when they rushed back to the kingdom, possibly out of concern for Olaf since people might get frightened by him and attempt to harm him.
Olaf also seems to consider Kristoff a good friend, despite the fact that during their first meeting, he called Kristoff "a funny looking donkey". For example, Olaf showed concern for Anna and Kristoff, and told them to run while he tried unsuccessfully to distract and hold off Marshmallow, indicating that Olaf cared for their safety and that Olaf considered Kristoff a good friend as he wanted to save him. He also told Anna that Kristoff loved Anna, and that his act of returning her to Hans to save her was an act of true love, showing that he understood Kristoff well enough to become aware of his feelings for Anna. However, Olaf did think Kristoff was crazy for a moment when Kristoff brought everyone to his family of "rocks", and warned Anna to run before his family of rocks revealed themselves to the group.
Although they don't interact much in the film, it would seem that Kristoff sees Olaf as a good friend and vice versa, which is made stronger by their equal love for Anna.
Although she had very little interaction with Kristoff in the film (including after she struck Anna in the heart with her powers and the ending), it is enough to show that although she was scared and wary to see him at first, as she was with pretty much everyone, she ultimately comes to accept him after knowing what he did for her sister. When Kristoff attempted to confront Hans for trying to kill the two sisters, Elsa seems touched by Kristoff's protectiveness over her and Anna. She even gives him the title "Official Arendelle Ice Master and Deliverer" so he can be with Anna.
It is also shown that Kristoff does care about Elsa to some extent, as he attempted to confront Hans for trying to kill her and Anna. It is worth noting that he was presumably the first person to learn about Elsa's powers aside from her family. In the film's novelization and coloring book, Kristoff saved Elsa by knocking out Hans.
In Frozen Fever, it is revealed that Kristoff and Elsa have developed a healthy friendship, with both expressing feelings of genuine comfort and acceptance with one another, and Kristoff serving as a sort of confidante to the Snow Queen.
When Kristoff and Sven were kids, they were orphans (what happened to their biological parents is unknown) until the Trolls took them in. Judging by how happy the trolls were when Kristoff and Sven came home, they obviously love the two, and think of them as family.
When he first hears of Hans, Kristoff questions Anna's judgment for agreeing to marry a man that she had just met. However, since he never personally meets Hans until the end, he has no actual opinion of him, although he does seem to find it suspicious that Hans would propose to Anna so quickly, questioning his character and how he views Anna. Nonetheless, Kristoff understands Anna's feelings for him, and for this reason can not understand (or possibly not want to admit) his feelings for her so she could be happy with Hans.
Later on, Kristoff brings Anna back to Hans so that he can cure her frozen heart, not knowing about Hans' true nature. This breaks his heart, but he believes Anna would be happier with Hans than with him. He appears to be aware of his betrayal, because when Elsa finishes thawing the kingdom, Kristoff looks about to confront Hans, but Anna stops him so she can do it herself (resulting in her punching Hans off the ship). In the original ending, it was planned that Hans would have a second wind and try to attack the sisters again while Elsa was cradling Anna's frozen body, only for Kristoff to knock him out. This was cut so that the focus was kept on Anna and Elsa where it belonged, rather than a battle between Hans and Kristoff, and so Anna would have the rightful honor of punching him (the scene was retained in the novelization).
Although the two have no actual interaction with each other during the film, (other than Kristoff nearly punching Hans after Arendelle is thawed), it is evident that Kristoff is aware of Hans' betrayal and attempt to kill both sisters and hates Hans, especially after he hurt and betrayed Anna, though Hans' view of Kristoff is unknown, though it can be presumed in the coloring book and novelization that he sees Kristoff as an impediment to his plans.