Describing the US leader as "a rogue and a gangster fond of playing with fire", Mr Kim said his counterpart was "unfit to hold the prerogative of supreme command of a country."
The statement carried by North's official Korean Central News Agency, comes after the US President gave a bellicose speech at the UN General Assembly in which he threatened to “totally destroy” North Korea if it did not step aside from its nuclear weapons programme.
“Rocket man is on a suicide mission for himself and his regime,” Mr Trump said. He said of the US: “If it is forced to defend itself or its allies, we will have no choice but to totally destroy North Korea.”
Mr Kim said Mr Trump's remarks have convinced him "that the path I chose is correct and that it is the one I have to follow to the last."
He added that he was "thinking hard" about his response, but Mr Trump will "pay dearly for his speech calling for totally destroying" North Korea."
Tensions between the two nations have ratcheted up in recent months as North Korea has continued nuclear and ballistic missile testing in a bid to give it the ability to target the United States with a powerful, nuclear-tipped missile.
The secretive communist state also claimed to have successfully detonated a nuclear weapon meant for an intercontinental ballistic missile and has claimed it has developed an advanced hydrogen bomb of "great destructive power".
An earthquake registering 6.3 was detected on the Richter Scale, shortly after the test earlier this month.
Mr Kim subsequently vowed to "establish the equilibrium of real force with the US and make the US rulers dare not talk about military option for the DPRK,” referring to North Korea‘s official name, the Democratic People’s Republic of Korea.
Mr Trump and Mr Kim have been testing each other for months now, and the President has employed increasingly harsh tones to discuss Mr Kim.
After North Korea threatened to launch a missile at the American territory Guam, Mr Trump promised that the US would bring "fire and fury" to North Korea if the country continued to threaten the US and its allies.
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