Speaking in the Oval Office, he said no other president should have to be investigated over "a false narrative".
He spoke a day after the attorney general released a summary of Special Counsel Robert Mueller's completed report.
It clears Mr Trump of conspiring with Russia with steal the US 2016 election.
But the long-awaited report stops short of exonerating Mr Trump of obstruction of justice.
US Attorney General William Barr ruled there was no evidence requiring prosecution on the obstruction issue.
What did President Trump say?
Mr Trump was hosting Israeli Prime Minister Benjamin Netanyahu in the White House on Monday when a reporter asked him about the outcome of the Mueller report.
"There's a lot of people out there that have done some very, very evil things, very bad things," Mr Trump said, "I would say treasonous things, against our country."
"And hopefully people that have done such harm to our country, we've gone through a period of really bad things happening.
"Those people will certainly be looked at, I've been looking at them for a long time.
"And I'm saying, 'why haven't they been looked at?' They lied to Congress - many of them, you know who they are - they've' done so many evil things."
Mr Trump did not name the alleged culprits.
He added: "It was a false narrative, it was terrible thing, we can never let this happen to another president again, I can tell you that. I say it very strongly."
What's the political reaction?
On Monday, Senate Judiciary chairman Lindsey Graham laid out the Republican strategy as he pledged to "unpack the other side of the story" of the Russia investigation.
The South Carolina senator, who spent the weekend with Mr Trump in Florida, said his panel would investigate the Department of Justice-led inquiry.
The FBI's use of a dossier compiled to discredit Mr Trump by a former British spy, Christopher Steele, would be among aspects under scrutiny, said Mr Graham.
Meanwhile, Capitol Hill Democrats are focusing on a line in the attorney general's summary that says Mr Mueller's report "does not exonerate" Mr Trump of obstruction of justice, even though Mr Barr concluded on Sunday there was insufficient evidence that Mr Trump had committed a crime.
House Judiciary Committee Chairman Jerrold Nadler said he would summon Mr Barr to testify soon "in light of the very concerning discrepancies and final decision making at the justice department".