During an interview with Fox News, Mr Giuliani was asked why he was involved, rather than the State Department or the attorney general.
“Why are you the president’s personal attorney?,” added Martha MacCallum. “What is your personal mission?”
“Wow,” Mr Giuliani responded before adding: ”To disrupt the world.”
He continued: “My mission is to defend my client in the best traditions of the legal profession.... It is in the best interests of my client to unravel the corruption in the Ukraine which involve mostly collusion in the election into which Joe Biden inserted himself.”
Mr Giuliani, who has been subpoenaed for documents by the House Democrats, has repeatedly claimed that Mr Biden acted corruptly in Ukraine on behalf of his son Hunter.
A former prosecutor in Ukraine said last month the the former vice president's son "did not violate anything".
It has since emerged that he was warned by US envoy Kurt Volker that his sources were “untrustworthy”. Mr Biden’s campaign has complained that Mr Giuliani is spreading “false, debunked conspiracy theories on behalf of Donald Trump”.
Mr Giuliani claimed his investigation concluded before Mr Biden announced he was running for president in 2020 – although the dates he gave suggested his investigation ended before it began.
“I started the investigation in November 2018, I ended the investigation in March 2018,” he said. “How could it possibly be to affect the 2020 election?
“How would I have known back then that he was going to run and why would I have not investigated him if he was central to the whole story?”
Several hours after the interview, Giuliani went online to accuse Democrat congresswoman Rashida Tlaib of suggesting that Mr Trump, his lawyers and other White House officials could be arrested by police.
In an apparently angry 4.54am tweet, he made the baseless claim it was ”evidence of a conspiracy to violate civil rights and to defraud us”.
“Failure to cover it by blaring headlines shows you’re part of Corrupt News Network,” he added.