I wonder how Vito felt...

Knowing that either Tessio or Clemenza was going to betray the family. He had been comrades-in-arms with them for up to 30 some odd years at that point.

Or does Vito view it as business, not personal.

In the novel, at least, Michael includes Carlo among the suspects. So maybe Vito figured Tessio and Clemenza would stay loyal to him, so that Carlo would the one again.

Business, I'm guessing. I think that betrayal was never far from Vito's realm of possibilities--comes with the territory. Attest his policy of giving orders to one person at a time, or through Tim, to prevent a corroborating witness who could testify against him. I think he also advised Michael to have everyone's phones tapped or monitored. Mostly, I think he'd be rightly suspicious that Michael wouldn't be accepted as his successor--he'd see that as business, not personal to him.
Nice to see you again, DT!

I think that one of Vito's strengths was the ability to separate personal feelings from business
He surely was aware that betrayal was part and parcel of the Business he had chosen.
He probably was disappointed that either Clem or Tess would turn on him,but wouldn't let it affect his course of action.
Michael made a couple references to things that Vito had probably taught him.
1) Keep your friends close,but your enemies closer.
2) Never hate your enemy,it clouds your judgement.

I'm willing to bet that Michael hoped that Carlo was the traitor. Puzo goes into Michael's thinking as well earlier in the book when it's time to decide whether Clemenza or Paulie set-up the Don. We know that Michael hoped it wasn't Clemenza because, well, it was Uncle Pete. Heck, it's not out of the realm of possibility that Clemenza or Tessio was Michael's Godfather? Clemenza is Santino's Godfather though, and being unfamiliar with Italian culture, is it common for someone to act as Godfather to more than 1 child from a particular family?

I reckon, in this instance, both business and personal

He was cold and businesslike telling Michael, Now listen -- whoever comes to you with this Barzini meeting -- he's the traitor. Don't forget that and personal that sadly his trusted Clem or Tess would be turning traitor for his son's murder

That's an important pount: Vito understood human nature. He saw how fear, anger, pride, greed, lust for revenge, blinded otherwise sen.sible men and led them to take damgerous risks and to make fatal mistakes. He controlled his emotions, tempered his responses, focused on results. As he put it in the novel: "Revenge is a dish best eaten cold."

My theory is that he always suspected it. They were loyal, even friends, but it was never 100% trust because it couldn't be. They weren't family, and outsiders can never be completely trusted (incidentally this is also why I thought Fredo's betrayal hurt Michael more than anything else; as family, Fredo SHOULD have been beyond reproach). So it probably came down to, as Michael felt as well, that it was business and the "smart move". It wouldn't have hurt any less (you can kind of tell that Michael is disappointed that he has to have Tessio killed, but it's business not personal. As Tessio even said before being led away) but it would have been necessary. It's the life they chose.