Kung is the 79th-generation descendant of Confucius in the main line of descent, making him the titular head of the Kong family. Kung succeeded his grandfather Kung Te-cheng to the post of Sacrificial Official to Confucius in 2009 following the latter's death a year earlier and was at the same time appointed Senior Advisor by President Ma Ying-jeou.
The 垂 Tsui character in his name is the generation name for 79th generation descendants of Confucius. His children all have the character 佑 Yu in their names, it being the generation name given to 80th generation descendants of Confucius.
His father Kung Wei-yi predeceased his grandfather. His mother was Kung Wei-yi's wife Yu Yuejie (于曰洁). Kung Tsui-chang and his wife, Wu Shuoyin (吳碩因) have a son Kung Yu-jen (孔佑仁), born 1 January 2006 in Taipei, and a daughter, Kung Yu-xin (孔佑心).