"The mentally deranged behavior of the U.S. president openly expressing on the UN arena the unethical will to 'totally destroy' a sovereign state, beyond the boundary of threats of regime change or overturn of social system, makes even those with normal thinking faculty think about discretion and composure," Kim said in a statement circulated on state news agency KCNA.
Kim urged the president to "exercise prudence in selecting words" after Trump on Tuesday issued a harsh warning to Pyongyang. The president threatened to "destroy" the communist nation if it threatened the United States.
Early Friday, Kim said that the president's remarks at the U.N. merit the "highest level of hard-line countermeasure in history." The dictator said he is "thinking hard" about what kind of reaction Trump could have possible expected when he delivered those remarks.
"Whatever Trump might have expected, he will face results beyond his expectation," the dictator said.
The North Korean leader concluded his statement by saying that he will "surely and definitely tame the mentally deranged U.S. dotard with fire." Kim also called Trump a "rogue and a gangster fond of playing with fire, rather than a politician."
Pyongyang's response comes hours after Trump signed an executive order that aims to expand his authority to target people and institutions doing business with North Korea. The measure seeks to cut off North Korea's access to funding and deter its nuclear and missile programs.
In recent weeks, North Korea has tested ballistic missiles and an apparent hydrogen bomb in the face of international economic sanctions and warnings. Last week, the U.N. unanimously passed fresh measures to punish the isolated nation economically, with the support of China and Russia.
Trump has repeatedly pressed China, North Korea's only major ally, to do more to force Pyongyang to abandon its nuclear ambitions.
On Tuesday, he also commended Beijing for signing on to two recent sanctions packages enacted by the Security Council. The U.S. sees China's commitment to sanctions as crucial to forcing Pyongyang to end its nuclear and missile programs.
Trump appeared to try to quash speculation that he is targeting China or other North Korean trading partners with the action.
"I want to be clear — the order targets only one country, and that country is North Korea," he said.
— CNBC's Jacob Pramuk contributed to this report.