I think commencement of the war was a two-step process. The first was Sollozzo-Tattaglia (perhaps with support from Barzini) v. the Corleones, over drugs. The second was the Five Families against the Corleones, which was set off by the killing of McCluskey. The Corleones were able to defuse McCluskey as an issue, but by that time blood had been spilled and everyone wanted a piece of the drugs action as part of the peace.