Ashley Hawkins’s body was found in the Scotland Street property on Friday after officers were called out to an early morning disturbance.
It has now emerged that the 32-year-old, who was named by police yesterday, had fled his home town of Barry, in Vale of Glamorgan, where he was wanted for burglary.
He was also suspected of stealing a vehicle in Cardiff.
Police issued a public appeal for information on his whereabouts last month.
A 28-year-old man from Wales appeared at Edinburgh Sheriff Court on Monday charged with Mr Hawkins’s murder.
Kieran Davies appeared in private and made no plea or declaration. He was remanded in custody.
In a statement released through Police Scotland yesterday, relatives paid tribute to Mr Hawkins.
They said: “Our beautiful boy Ashley, we can’t believe that he is gone.
“He was a loving son, brother and uncle and our hearts are now in pieces.
“He was helpful to everyone and was loved in his hometown of Barry. He will never be forgotten.”
Police made a fresh appeal for information on the death.
Detective Inspector Stuart Alexander of the Major Investigation Team said: “This is a tragic incident and our thoughts are with Ashley’s family at this time. We are still conducting inquiries in connection with Ashley’s death.
“I would urge anyone with information which may be able to help, or who was in the Scotland Street area early on the morning of Friday, December 2, to come forward.”
Residents of Scotland Street, home to some of the Capital’s most expensive properties, told the News last week of their horror at the discovery of a body.
A full major investigation team, involving up to 45 police officers and staff, was working on the case.
Officers forced their way in to the flat on Friday morning after receiving calls from several neighbours about a serious disturbance.
Two vehicles in the street were taped off later the same morning
Those with information can contact Police Scotland on 101 or anonymously through the charity Crimestoppers on 0800 555 111.