The US president said on Tuesday that if America was forced to defend itself it would "totally destroy" North Korea.
Mr Trump also mockingly called Mr Kim a "rocket man" on a "suicide mission".
The two countries have been engaging in increasingly heated rhetoric in recent months.
North Korea has been testing missiles at an unprecedented rate, and conducted its sixth nuclear test despite international condemnation.
North Korean Foreign Minister Ri Yong-ho, who had earlier compared Mr Trump's speech to "the sound of a barking dog", has warned that Pyongyang could test a hydrogen bomb in the Pacific Ocean in response to the US president's threat.
Mr Kim said in an English statement carried by state news agency KCNA that Mr Trump's remarks "have convinced me, rather than frightening or stopping me, that the path I chose is correct and that it is the one I have to follow to the last".
He said "now that Trump has denied the existence of and insulted me and my country in front of the eyes of the world and made the most ferocious declaration of a war in history", North Korea would consider the "highest level of hard-line countermeasure" to make Mr Trump "pay dearly for his speech".
Media captionTrump: 'Rocket Man's suicide mission'
He ended by saying he would "surely and definitely tame the mentally deranged US dotard with fire".
Experts say this is the first time a North Korean leader has made a direct address to an international audience.
The statement came shortly after the country's delegation arrived in New York for the UN General Assembly. Analysts say that for this reason it merits serious and thorough consideration.
Its first known use was in the 14th Century and initially meant "imbecile", according to the US Merriam-Webster Dictionary. It stems from the Middle English word "doten" which means "to dote"
Searches for the term have been "high as a kite" since Mr Kim released his statement, Merriam-Webster tweeted
Mr Kim's comments prompted swift criticism from the Japanese government.
Chief Cabinet Secretary Yoshihide Suga said in a news conference on Friday: "North Korea's remarks and behaviour are provocative to regional and international security, and they are absolutely unacceptable."
North Korea has fired two ballistic missiles over Japan in the past month, raising regional tensions even further.
Mr Trump on Thursday signed a new order boosting sanctions against North Korea, where the US treasury would target firms and financial institutions doing business with Pyongyang.
He said: "For much too long North Korea has been allowed to abuse the international financial system to facilitate funding for its nuclear weapons and missile programs."
The UN Security Council had approved new rounds of sanctions earlier this month aimed of starving North Korea of fuel and income, which were in response to Pyongyang's much-condemned sixth nuclear test on 3 September.
Media captionHow would war with North Korea unfold?
North Korea was one of the top issues dominating the agenda at the UN General Assembly in New York, where several speakers called for a de-escalation of tensions.
Russian Foreign Minister Sergei Lavrov warned that "military hysteria" over the nuclear testing would lead to "disaster", while China's Foreign Minister Wang Yi told Pyongyang not to go in a "dangerous direction".
Mr Wang told the UN on Thursday that there should be no new nuclear weapons on the Korean peninsula "whether it is in the North or the South".
