As far as I know, there is no official chairman of the Commission, although an individual Don might be a kind of de facto chairman by force of personality--Carlo Gambino pretty much fulfilled that role. The only influence he'd have as de facto chairman would, again, be through force of personality and persuasiveness, rather than by any "rules" of the Commission (if there are any).

As you allude to, Charlie Luciano created the Commission in part to eliminate the struggle over who was the "boss of bosses" that resulted in the Castellemmarese War and the murders of Joe Masseria and Salvatore Maranzano. Luciano didn't have to appoint himself chairman--everyone knew he was first among equals because of his pivotal role in the War, and in creating the Commission.