Born on New Year's Eve in 1977 in the Livry-Gargan suburb of north-eastern Paris, Cheurfi was jailed four times for attempted murder, violence and theft.
Cheurfi had four criminal convictions, the prosecutor said. The killer had spoken about wanting to kill police officers but had had no apparent link to Islamist radicalism, Mr Molins said.
Cheurfi was arrested and his home searched in February after an inquiry was opened over reports that he had been talking of killing police and getting weapons, Mr Molins said.
Hunting knives, masks and a GoPro camera were found but this was not considered sufficient proof of his murderous intent.
In April 2001, while driving a stolen car, he fired on and wounded two men who had chased him, one of them a plain clothes police officer.
He last emerged from prison in October 2015 and was living in the eastern suburb of Chelles.
His lawyer described him as psychologically fragile and neighbours in Chelles said he had never shown any interest in religion.
Three members of the gunman's family have been arrested for questioning.
Driving an Audi, the gunman arrived in the Champs Elysees area at 20:47 (18:47 GMT) on Thursday and drove up to a police bus parked on the avenue, Mr Molins said.
Getting out, he opened fire at the driver's window, killing a police officer inside. The officer has been named as Xavier Jugelé.
He used a Kalashnikov assault rifle to kill the police officer with two bullets to the head, François Molins told journalists in the French capital.
A note defending so-called Islamic State was found near his body.
He then went around the bus and fired on two police officers posted outside a Turkish tourist office. It was announced earlier that two officers had been injured.
Security forces later shot Cheurfi dead.
Source: rt.com, news.lv, bbc.co.uk