"Do not underestimate us. Do not try us," he said, while also condemning the "dark fantasy" of life in the North.
He addressed Mr Kim in his speech, saying "the weapons you are acquiring are not making you safer, they are putting your regime in grave danger".
The US leader is on a two-day visit to South Korea as part of an Asian tour.
Mr Trump began his speech to lawmakers in Seoul by praising the South's achievements since the Korean War, in contrast to the North's. He likened it to "a tragic experiment in the laboratory of history".
He said North Korea was ruled by a "military cult" with the "deranged belief in the leaders' destiny to rule as parent-protector over a conquered Korean peninsula and an enslaved Korean people."
He warned Pyongyang that while it may have interpreted past restraint by the US as weakness, there was now a different US administration in place.
Mr Trump also called on the rest of the world - while singling out China and Russia in particular - to exert pressure on Pyongyang to give up its nuclear weapons.
"The world cannot tolerate the menace of a rogue regime that threatens it with nuclear devastation. All responsible nations must join forces to isolate the brutal regime of North Korea, to deny it any form of support, supply, or acceptance," he said.