Kim and Trump vowed to work toward denuclearisation and build a "lasting and stable" peace regime at their landmark summit in Singapore in June, but little progress has been made since and satellite images have indicated continued activity at North Korea's nuclear and missile facilities.
In a speech broadcast by the North's state television, he said there would be faster progress on denuclearisation if Washington took corresponding action.
But for most of his 30-minute speech, he focused on North Korea's economy and said he stands ready to resume projects with South Korea and restart operations at a jointly run factory park in the border town of Kaesong.
The comments are likely to fuel growing scepticism over whether Pyongyang intends to give up the nuclear weapons program that it has long considered essential to its security.
Talks between the USA and North Korea have made little progress since Trump and Kim held an unprecedented summit in June and signed to "work toward complete denuclearization of the Korean Peninsula", without defining the term or setting a deadline.
Seoul and Washington are in a security alliance and the U.S. stations 28,500 troops in the South to protect it against its neighbour, which invaded in 1950.
Kim's speech comes after year of diplomatic surprises by the reclusive leader, including an unprecedented summit with Trump and three meetings with Moon.
The leader's New Year speech is a key moment in the North Korean political calendar, reviewing the past and setting out out goals for the future.
Pyongyang conducted its latest missile launch in November past year.
Observers say those steps have not been confirmed and can be easily reversed.
"Now that North and South Korea chose to take the path of peace and prosperity, we demand that joint military exercises with outside forces should no longer be allowed and deployment of war equipment such as foreign strategic assets should be completely stopped", Kim said.
"We will never tolerate outside interferences and interventions meant to block our way toward reconciliation, unity and the unification of our people, while trying to make our relations scummy to their states and interests", the North Korean leader said, adding that "significant" agreements achieved during 2018's three inter-Korean summits could be "regarded as non-aggression treaty".
In a jab at the South's alliance with the U.S., Kim also said Washington should continue to halt its joint military exercises with Seoul and not deploy strategic military assets to the South.
Kim also reiterated his "strong will" and the North Korean government's "unchanged" position to pursue better relations with the U.S. and to seek complete denuclearization.
As a way to tackle chronic power shortages, he raised the need for atomic energy.
Among the topics the leader is expected to address Tuesday are the state - and perhaps future - of denuclearization talks with the US, relations with South Korea and the Hermit Kingdom's economy, which has been tanked by sanctions from the USA and worldwide community after years of nuke tests. Kim also urged the North Korean people to strengthen their "self-reliance", a message that echoed Chinese President Xi Jinping's similar address Monday night.Kim repeated the importance of being "frugal" multiple times when speaking about economic plans.