He put the rest of the remains in a wheelie bin and left them at the property for another eight months before dumping them in the garage of the property next door, which had been abandoned. The remains were eventually found when a demolition company went to work at the property.
The Crown alleges Cooper then posed as Mr Mills, sending his family and friends messages via social networking site Bebo, in a bid to lead them to believe he had moved to Wellington to start a new life.
He also used Mr Mills' Eftpos card, withdrawing his Work and Income New Zealand benefit each week and making purchases.
Ms Pollett said the destruction of Mr Mills' skull was highly relevant to the Crown case.
She said the skull was a key piece of evidence, "crucial in establishing the cause of death. The absence of the skull unlocks the door to the truth that the accused does not tell us", she told the jury.
"The skull had to be destroyed for fit with the theories (Cooper) wished to advance."
Ms Pollett told the jury they would see photographs of three crime scenes - the address where Mr Mills was killed and first buried, where his body was later dumped and a third property where bone fragments were found.
She acknowledged parts of the case would be gory, and urged jurors not to dwell on the unpleasantness of what they were seeing.
Cooper became angry with Mr Mills, according to the Crown, after they began living together. Cooper was "incensed" with Mr Mills, who owed him money, would eat his food, use his power and wear his clothes.
That anger eventually led to the altercation in which Mr Mills died.
Cooper has not denied killing Mr Mills but says the death was accidental and has denied the murder charge.
He earlier pleaded guilty to covering up the death, and that information was shared with the jury today.
None of the evidence about his attempts to make people believe Mr Mills was still alive is disputed by the defence.
Defence lawyer Mina Wharepouri also made a brief opening statement and said Mr Mills' death was a tragic and unforeseen accident.
He argued Mr Mills tried to attack Cooper, who used "reasonable force" to defend himself.
What he did next was "cowardly", but it did not make him guilty of murder.
"The facts of the case are macabre, but that is not murder," he said.
Justice Allan told jurors they must approach their task "without the element of sympathy" and "without the introduction of any emotion".
He also reminded them not to do their own research on the case outside court by reading media reports past and present.
"You must decide purely on the basis of whether the Crown has proved its case," he said.
"You are the sole judges of fact... It's for the Crown to prove beyond reasonable doubt that Mr Cooper was responsible for the murder of the deceased.
"It's not for the defence to prove that Mr Cooper is not guilty."
Cooper did not show any emotion in court as he sat in the dock flanked by a security guard.
His expression did not change as Ms Pollett outlined the allegations against him, including the grisly details of Mr Mills violent death and the various attempts to dispose his body.
The trial will continue tomorrow, when it is expected the jury will hear from the first of 56 witnesses including friends and family of both Cooper and Mr Mills.
