Simon Wishart, 22, claimed the victim had committed suicide but later allegedly confessed to Lewes Prison staff that he had killed him while warbling.
Wishart is accused of throttling 25-year-old Gareth Russell who was found dead in cell 19 at the Brighton Road jail's care and support unit for vulnerable prisoners.
The killing was allegedly carried out just after midnight on April 24 last year.
Lewes Crown Court heard that Mr Russell had been convicted of manslaughter just days earlier and was starting a six-year jail sentence.
When Wishart first raised the alarm he claimed he had found Mr Russell hanging from heating pipes by a ligature made from torn bed sheets, the court was told.
But Wishart, who was in jail on remand awaiting trial for robbery, allegedly later admitted killing him.
He is said to have told a member of prison staff that while he killed his cellmate he started singing the American national anthem because Mr Russell would 'wind him up' as Wishart was originally from the US.
Prosecutor Jeremy Gold, QC, told the court, "The prosecution case is the defendant deliberately strangled Gareth Russell with a ligature before embarking on a bizarre attempt to make it look like Mr Russell had hanged himself in his cell.
"The murder of Gareth Russell was carefully planned by someone who knew exactly what he wanted to achieve."
Mr Gold said that when police investigated the apparent suicide they discovered Wishart and Mr Russell had played games together.
He said these involved burning each other with cigarette lighters to relieve the boredom of being in jail.
Mr Gold said the jury would hear evidence from other prisoners who claimed Wishart had threatened to kill Mr Russell because he annoyed him.
The day before the alleged murder he is accused of telling another prisoner he was going to kill Mr Russell because he was annoying him.
On the day Mr Russell died, Wishart is said to have told his girlfriend that his cellmate was getting on his nerves.
Just after midnight, Wishart rang his cell bell and told the prison officer who answered, "Get me out of here. He has hung himself."
Mr Gold said Mr Russell's body was on the floor and he was clearly dead. A ligature made from strips of bed sheet was around his neck.
Three days later Wishart was arrested and made comments to officers guarding him overnight at the police station.
He allegedly said, "Maybe I did do it, maybe I didn't. Maybe I will never know. Before I die I want to be known."
The court heard Wishart returned to jail and was sent to the hospital wing where he confessed to a member of staff that he had strangled Mr Russell and tried to make it look like suicide.
He claimed he had killed his cellmate because Mr Russell had touched him sexually when they were tied up during one of the games they played together.
It was then he made the claim about singing the American national anthem, the jury was told.
He later claimed Mr Russell died by accident after seeing how long they could hold their breath while something was tied around their necks.
Wishart denies murder.
The trial continues.