Tony Thao Do is accused of killing Kim Lien Huynh, 54, in her Cabramatta home on September 13, 2012. Mr Do has pleaded not guilty, saying someone else was responsible for the attack.

On Thursday, a NSW Supreme Court jury heard Mr Do and Ms Huynh had met online in either 2006 or 2007 and shared a passion for internet karaoke and the online video chat site Paltalk.

Mr Do eventually moved to Australia in September 2011 and began living with Ms Huynh and her adult daughter Aminda.

Aminda, a physiotherapist, told the jury the couple were inseparable at first, with Mr Do helping in Ms Huynh's strata management business at Canley Vale.

But about four months before her death, they stopped sleeping in the same bed and had regular arguments about how much time Mr Do spent on the computer and Ms Huynh's wish for him to sign a prenuptial agreement.

"She was angry at him for not doing more around the house or getting more education," Aminda said.

"She would ask us [Aminda and her siblings] whether he should go back to the US, whether she had made the right decision and what we thought about him."

In cross-examination by defence barrister Craig Smith, SC, Aminda agreed her mother had told Mr Do: "You are so lazy, why are you here, I want you to go back to America."

Aminda also described how, on the night of her mother's death, she walked through her house after work, made chicken pies for dinner, watched TV, and sent texts to friends.

As she could see her way around the Alick Street house in the glow from the fish tank, so she did not switch on any lights until after midnight, when one of her dogs needed to go outside.

It was only then that she saw her mother lying bloodied in the front room of the house, cold to the touch.

A post-mortem examination showed Ms Huynh died from the combined effects of head injuries and "neck compression".

The trial before Justice David Davies continues.