Kim is wanted in Korea in relation to the murder of his mother, stepfather and 14-year-old half brother on October 21, Korean media reported.
South Korean police said the bodies of Kim's mother and son were found with stab wounds in an apartment, and his stepfather's body was found in the boot of a car.
The arrest is the first step to allowing authorities to look at an extradition.
Lawyer Joon Yi, who is acting for Kim, told the Herald he will not oppose an extradition.
Discussions are under way with the authorities to start the extradition process, Yi said.
"It is all just a matter of paperwork now," Yi said.
"Mr Kim is due to reappear on December 1, but if the extradition can be organised earlier then he will be gone."
When asked how long he thought the process would take, Yi said: "That's what my client wants to know too."
Prosecutor Emma Hopkin said in court that Kim met all the criteria for extradition.
The allegations he faced in Korea were of the "utmost seriousness", she said, and Kim had also shown a willingness to travel.
Kim was arrested in Auckland on Sunday evening in a building while his wife was in a car.
It was revealed that an eagle-eyed local Korean community member helped catch Kim.
The would-be sleuth alerted the Korean Consulate in Auckland after spotting the suspect moving into an Auckland home last week.
He recognised Kim from Korean social media reports about the murders.
He later contacted Korean police who are believed to have alerted their New Zealand counterparts, leading to the suspect's arrest.
Kim, who is a New Zealand permanent resident, reportedly flew to New Zealand with his wife and two young daughters, arriving in Auckland on Tuesday last week.
Details of the case, the suspect's name and a car he purchased in New Zealand were discussed widely and circulated on Korean social media.
The Auckland-based Korean man who thought he recognised Kim approached the consulate to report his whereabouts, Korean media have reported.
The consulate confirmed to the Herald it had been contacted by the informant, adding that its staff had worked tirelessly with the Korean Embassy and local authorities since the story broke.
The informant had seen Kim with a luxury car that fitted the online description.
He thought it was suspicious that the man was moving into a "huge mansion" without any luggage.
The suspect had also ordered a fridge, washing machine and television to be delivered on the same day.
After approaching the consulate in Auckland, the informant then called Yongin Dongbu Police after searching for their contact details online.
Kim pleaded guilty to a charge of theft yesterday and was further remanded in custody till next month.
"We are satisfied with the level of co-operation with the Republic of Korean Consulate in Auckland," a police spokeswoman said.
"Police [will] continue to liaise with Korean authorities regarding this matter."
A Korean Consulate spokeswoman said: "The consulate has been working tirelessly with the Korean Embassy in Wellington from the outset of the incident and will continue liaising between the South Korean Government and Crown Law in a timely fashion in this matter."
