I think Barzini's moves against Vito were "personal" in the sense that he thought of himself as No.1 waiting in the wings, impatient to be the top dog in NYC, younger than the other Dons, whom he regarded as "Moustache Petes" (per the novel). But, I don't think he had a personal grudge against Vito.

I also don't think he was in a big hurry. He waited until after Vito died to make his move against Michael. That was years after he and Vito "made the peace" after Sonny died.