Police dashcam video shows Danny Ray Thomas, 34, walking towards the officer on a road in Houston, Texas, last Thursday before a gunshot rings out.
Deputy Cameron Brewer repeatedly shouts “get down on the ground” before firing. Both the officer and victim are black.
Mr Thomas’ family say he suffered from depression after his children died.
His sister, Kita Thomas-Smith, attended a Houston City Council meeting on Tuesday where she tearfully urged politicians do to something about “police brutality”.
Mayor Sylvester Turner told her: “I certainly feel your pain. We are certainly sorry for any life that is taken.”
Outside city hall, Ms Thomas-Smith told the Houston Chronicle of the dashcam video. “It feels like premeditated murder.
“He was clearly walking, not running, toward the deputy not trying to hurt or harm him.”
Deputy Brewer, who had a Taser with him, has been placed on administrative leave as police investigate the shooting.