In an unprecedented personal statement, via state media, Mr Kim said Mr Trump would "pay dearly" for his recent speech to the UN.
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.
"It could be the most powerful detonation of an H-bomb in the Pacific," Mr Ri said,quoted by South Korea's Yonhap news agency.
However, he added: "We have no idea about what actions could be taken as it will be ordered by leader Kim Jong-un."