yes sollozzo tried to kill the don becuase he don't want to share his political powers.
But Sollozzo wasn't a head of on of the New York five families. He had no family but he get the support of Tattaglia and Barzini. Sollozzo was just a narcotics man.
And the war was between the Corleone family who was against drugs and sollozzo and Barzini, Tattglia who want drugs and sollozzo. Cuneo and Strachi (the other 2 familys) just take side of Barzini and Tattaglia.