The first day of Eklund's trial at Stockholm district court on Monday was dedicated to the murder and rape of Pernilla Hellgren in Falun in 2000. Eklund admitted to raping Hellgren before killing her.
Anders Eklund had previously admitted to both killings when he was presented with DNA evidence from the Hellgren case.
Eklund appeared unmoved when the prosecutor detailed the injuries on Pernilla Hellgen's body when she was found. He initially claimed that Hellgren had consented to having sex with him. Eklund later confessed that Hellgren had tried to flee and that the sex was not consensual.
"I grabbed her from behind or maybe I pushed her over."
He asked for forgiveness for the murder.
"She was not supposed to die," Eklund said before bursting into tears.
"Forgive me," he repeated.
Eklund related how everything went wrong.
"There was no logic. There is a kind Anders and an evil Anders."
Hellgren's father Christer Lundin attended court proceedings on Monday morning.
"It is a relief that it gets going so that we can seek closure and that Pernilla can rest in peace. The last few days have been emotional."
When asked by news agency TT how it felt to read the newspapers, Lundin replied:
"It is horrible to see new stories every day about what he did to my daughter. It has been enormously emotional."
Anders Eklund has not himself read the newspapers in prison. But he has seen TV coverage, said Eklund's lawyer Leif Silbersky.
"He understands the enormous pressure he will be subjected to. I have tried to prepare him for the onslaught," said Silbersky.
He argues that Eklund can not be convicted of raping neither Pernilla nor Engla if he is he also convicted of murder.
"The murder consumes the rapes. Jurisprudence dictates that one can not be convicted of both," said Silbersky.
The trial is expected to continue for four days which is being held in the secure room of Stockholm district court due to threats directed at Eklund.
Engla's mother Carina Högglund will attend court together with the girl's step-father when her murder is considered by the court on Tuesday.
"They will be present at the indictment. The plaintiff can do so if, for example, they would want to express an opinion about the penalty to be served on the accused," said the family's counsel Catharina Wikner to news agency TT.
Anders Eklund stands accused of the aggravated rape and murder of 31-year-old Pernilla Hellgren in Falun in 2000 and 10-year-old Engla Juncosa Höglund in Stjärnsund in April 2008.