Three women and two men were found dead inside the home near City College on March 23.
The victims are a mother, father, adult daughter and son, and his girlfriend.
Police have not disclosed a motive, though sources say it had to do with gambling debts.
Binh Thai Luc, 35, was arrested two days after the killings.
He has pleaded not guilty to the charges and is expected back in court May 3.