As Michael took over from his father, he needed to determine if Tessio and Clemenza were going to be loyal to him. This was especially important because, as Vito stepped out of his role and then died, Barzini got more confident, more assertive, more sure that he could supplant the Corleones and become the Number One Mafia family in NYC. Michael was in danger. To test Clemenza and Tessio, he deliberately put on a show of weakness and diffidence. Clemenza stayed loyal, Tessio thought Michael would never be able to stand up to Barzini, so he betrayed Michael, thinking Barzini would then make him (Tessio) head of the Corleones. Michael could never give a traitor a pass, so Tessio was killed.