Bidhya Sagar Das, 33, was charged after the children were found with critical injuries at a flat near Finsbury Park, north London, on Saturday night.
The boy died in the early hours of Sunday and the girl remains in a critical condition in hospital.
Mr Das, from Hackney, was remanded in custody by Thames magistrates.
He will next appear at the Old Bailey on Thursday.
A post-mortem examination in relation to the boy is due to begin later. Formal identification has yet to take place.
The Metropolitan Police said detectives from the Homicide and Major Crime Command continue to investigate.
A man has appeared in court charged with the murder of a one-year-old boy and the attempted murder of a girl, also aged one.