Ashley Madison’s Canadian parent company Avid Life Media is offering a C$500,000 (£240,000) reward for information on the hackers, they added.
Details of more than 32m accounts were stolen from the website, which offers users the chance to have an affair.
Two individuals associated with the leak of Ashley Madison customer details are reported to have taken their lives, according to police in Canada.
The police in Toronto gave no further information about the deaths.
Addressing the hackers, known as The Impact Team, acting staff superintendent Bryce Evans of the Toronto police said: “I want to make it very clear to you your actions are illegal and we will not be tolerating them. This is your wake-up call.”
Police are seeking information from members of the wider hacker community that might aid their investigation.
The breach was “very sophisticated”, said Detective Menard from the technological crime unit of Toronto Police.
Mr Evans confirmed that credit card data was included in the original data dump released by The Impact Team.
He said that investigators believed this was limited to the last four digits of the main card number.
Consequently, police are advising victims of the hack to review their accounts.
He also explained that the hack had already led to a series of “spin-offs of crimes and further victimisation”.
“Criminals have already engaged in online scams by claiming to provide access to the leaked websites,” he said.
“The public needs to be aware that by clicking on these links you are exposing your computers to malware, spyware, adware and viruses.”
The unfolding of the hack was also detailed at the conference – from the moment on 12 July when several Avid Life Media employees logged in to their computers and were confronted by a message from the hackers.
This message was accompanied by music – AC/DC’s “Thunderstruck”, said Mr Evans.
“This hack is one of the largest data breaches in the world and is very unique on its own in that it exposed tens of millions of people’s personal information,” he added.
Police have set up a Twitter account, @AMCaseTPS, and hashtag, #AMCaseTPS, in a bid to gather information about the hack from members of the public.
In a statement, Avid Life Media confirmed the offer of a financial reward for information and it believed the investigation was proceeding positively.
“We are confident that the considerable investigative and prosecutorial power that is being brought to bear on this unprecedented crime will lead to arrests and convictions,” the firm said.