The singer, whose real name was David Emmanuel, died at his home in Surrey on Tuesday morning following a police raid.
His family have been told he stabbed himself while making a cup of tea in his kitchen while police were still inside his home.
Smiley Culture, 48, found fame with a string of 1980s' hits including Cockney Translation and appeared on Top of the Pops.
His family are now demanding an independent inquiry into the circumstances surrounding his death.
His relative Merlin Emmanuel told a packed press conference in Brixton, south London today: "The police have yet to give an official statement as to what happened.
"We haven't had a clear, coherent, official explanation as to what happened to Smiley.
"The police have a lot to answer to. Until our questions, queries and suspicions have been fully and competently answered to dispel any notion of foul play, we will not rest.
"The truth must be known. Justice must be served.
"Fact - Smiley Culture died at his home from a single stab wound whilst in police custody while they let him go and make a cup of tea."
Police watchdog the IPCC has launched an investigation into the death.
Merlin Emmanuel said the family had been inundated with thousands of messages of support from across the world.
He added: "On behalf of the Emmanuel family I would like to express our deep sorrow and anger about the events that led to the death of our beloved David Emmanuel.
"As a family we are in a state of shock and confusion and we are determined to search for the truth of what happened on the morning of March 15 2011, a day that will now be forever etched in our minds.
"My uncle David Emmanuel is now dead. He was a father, an uncle, a friend and a mentor to many and is a British icon who died under the most peculiar of circumstances. Smiley was the first recognised British rapper."
In September last year he appeared before magistrates charged with conspiracy to supply cocaine and was due to appear in court in the coming days.
But Mr Emmanuel explained: "Smiley had a great deal to look forward to. It is true that he had an impending case but it is also true that the evidence in relation to this case was minimal and he along with his legal team were supremely confident of being acquitted this coming week."
Family spokesman and race equality campaigner Lee Jasper said: "I've never heard the kind of scenario that was explained to me in relation to Smiley Culture's death.
"Even if you were to accept that it was true it would still be bizarre, so given that there is very little trust and confidence in the police it's no wonder that the vast majority of people who hear this tale are struck down with disbelief and anger."
Smiley Culture's 1984 hit single Police Officer was an autobiographical song of how he was caught in possession of cannabis but let off when the officer recognised him as a reggae artist.
Speaking about the hit, he told the Guardian last year: "My hit single Police Officer was a true story - the police used to take my weed. It was better than being arrested, and I made that into a hit."
Dancehall MC Asher Senator, a long-time friend, said: "Culture was original, there was no one like him, his courage was extreme.
"He was very confident and very clever. People would be talking and Culture would be thinking about next week, while we are thinking about £5, Culture's thinking about £5 million."
The IPCC confirmed today that the singer died of a single stab wound to the heart.
Outlining the details of the raid, in IPCC spokesman said: "The facts that have been established so far are that, at approximately 7am on Tuesday 15 March 2011, four MPS officers went to Mr Emmanuel's house on Hillbury Road in Warlingham, Surrey, to execute a search warrant.
"Later the same morning, at approximately 8.30am, an ambulance was called to attend the address by the MPS officers. An air ambulance crew arrived but Mr Emmanuel died at the scene.
"The case was referred to the IPCC and an independent investigation was launched."
Forensic experts are examining items which have been recovered from the address, including the knife.
IPCC Commissioner Mike Franklin said: "I send my condolences to Mr Emmanuel's family and friends. I understand their concerns and am under no illusion about the seriousness of this case, its impact on them and the wider community."
He said the investigation was examining the basis on which MPS officers went to Surrey to carry out the search warrant, how that was planned, how it was communicated with Surrey Police and what happened while officers were at the address.
The investigation will also focus on the response of the officers present to examine what was done in terms of first aid.
Mr Franklin added: "Many of the questions that I know Mr Emmanuel's family and friends have are exactly what we will be examining as part of this investigation.
"I want to reassure people that we will thoroughly scrutinise this incident and will update all those involved as we progress.
"I will also make the findings of our investigation public at the earliest opportunity. Until then, I would ask that speculation is avoided."
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