The US President told supporters America "really has no choice" but to confront the Pyongyang regime over its nuclear weapon ambitions.
And he pointed the finger at his predecessor, Barack Obama, and former secretary of state Hillary Clinton for not dealing with the North Korea threat earlier.
He said: "We can't have madmen out there shooting rockets all over the place.
"The 'Rocket Man' should have been handled a long time ago. He should have been handled a long time ago by Clinton... by Obama.
"This is a different time. This shouldn't be handled now, but I'm going to handle it, because we have to handle it."
On Thursday, Mr Kim said Mr Trump would "pay dearly" for a threat he made during a speech at the United Nations General Assembly.
In that speech, the US President mocked the North Korea leader as a "Rocket Man" on a "suicide mission".
He added that if "forced to defend itself or its allies, we will have no choice but to totally destroy North Korea".
Mr Kim had insisted his country's nuclear programme will continue, hinting it may explode a hydrogen bomb, and vowed that Pyongyang would consider the "highest level of hardline countermeasure in history" against the US.
He described Mr Trump as "mentally deranged", adding: "After taking office, Trump has rendered the world restless through threats and blackmail against all countries in the world.
"His is unfit to hold the prerogative of supreme command of a country, and he is surely a rogue and a gangster fond of playing with fire, rather than a politician."
Moscow's foreign minister Sergei Lavrov has described the scrap as resembling a "kindergarten fight between children".
"We have to calm down the hot heads and understand that we do need pauses, that we do need some contacts," Mr Lavrov told a news conference after his address to the UN General Assembly.
It comes after EU ambassadors agreed draft sanctions against North Korea, including a ban on investments in the country and on EU exports of oil.
China, the North's main trading partner, has also announced it will limit oil supplies to the reclusive state starting on 1 October.
Mr Trump signed an executive order on Thursday to introduce fresh sanctions on Pyongyang over its nuclear weapon programme.
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