The high profile murderer and sword attacker died in Auckland's Paremoremo Prison overnight.
A prison source told NZPA Dixon had suffered head and neck injuries when he struck his head on his cell bars.
Mr Hart said his client had refused to take his medication and had been beaten up several times by inmates.
He believed his client should have been put into a psychiatric unit, not a prison cell, he told Newstalk ZB.
Dixon was jailed for a 2003 P-fuelled rampage when he attacked two women near Thames with a samurai sword, severing or partly severing their hands.
He then drove to Auckland and killed James Te Aute by shooting him repeatedly in the back.
The Corrections Department confirmed Dixon was found dead in his cell last night.
"Staff provided immediate medical assistance but were unable to revive him," Corrections assistant general operations Leanne Field said in a statement.
"He was pronounced dead by ambulance staff when they arrived at the prison."
An initial investigation to determine what had happened was under way.
"However, I can say that it appears the prisoner died of self-inflicted injuries," Ms Field said.
The death would be investigated by the Prison Inspectorate, who act independently of Prison Services, she said.
The investigation would be monitored by the Office of the Ombudsmen.
The announcement of his death came as Mr Hart and the Crown were due at the High Court at Auckland to make preparations for his sentencing, after a retrial last year.
Dixon was convicted in 2005 on charges including murder and kidnapping, receiving a 20-year minimum sentence, but that was appealed on insanity grounds and a re-trial granted.
At that five-week retrial in June last year Dixon was again convicted on several charges including murder and was in custody awaiting sentence.
Details about his upbringing emerged during his trials, including alleged sexual abuse on him as a child and incidents of him being kept outside tethered to a clothesline.
His defence argued he was insane, but the Crown countered that he was "not mad, but bad".
Mr Hart said in December last year he planned to appeal the second conviction and try to get a third trial.
He was vague about the grounds he planned to base the case on.
Dixon has managed to maintain a profile while in prison.
He was recently understood to have tried to pull a makeshift weapon on Mr Hart during a prison visit.
In December 2007 he used a fork to try to gouge out the eye of a fellow prisoner, who required hospitalisation.
It was later discovered Dixon had his own MySpace web page with comments about his crimes and photos of him in prison on it.
