How did Don Corleone know that Barzini was behind Sollozzo?

I've always thought that it was two things that lead Don Corleone to that realization:

1. As the Don himself puts it: "Tataglia's a pimp. He could never have outfought Santino". Obviously the Don had little respect for Tataglia's abilities as a war time Don. It's also obvious that it was a thought that had bothered him for quite some time.

2. During the meeting, it was obvious that Don Barzini took a very keen interest in the peace negotiations. In fact, he took almost complete reins of negotiating with the Corleones for a settlement. None of the other families seemed bothered by this. These facts indicate that the other families had already agreed on giving Barzini the lead role.

Now given how weak Tataglia was, and how strong Barzini was, it's not far fetched to deduce who was the master and who was the puppet.

When Sollozzo entered the picture, he requested the Corleone for protection and to give up political influence, something that he did not need at large for his business - so he was pushing his and someone else's agenda -, in return he will give a cut of the profits, and take care of the Tataglia from his end. Thus at 'The Peace Meeting', A meeting to discuss peace between the Tataglias and the Corleones, there was no much talk about the family feud from anyone other than from Don Vito and Tataglia, instead Barzini took that opportunity to (once again) 1)request for the Corleone to give up judges and political influence and 2)push the Sollozzo business (trafficking and dsitribution of drugs), as he said: "Don Corleone has all the judges and the politicians in New York, and he must share them, he must let us draw the water from the well". And went on to say: "Traffic and drugs will be permitted but controlled, and Don Corleone will give up protection in the East and there will be the peace". Barzini set then the peace terms, he obtained what he wanted from that meeting, and Don Corleone had to give up a lot for relative peace and to have Michael returned safely, that is when the Don realized that the one orchestrating everything from the beginning was Barzini.

In my opinion Barzini said, "A refusal is NOT the act of a friend." To me, that said that Barzini was interested in them making peace so that the drug trade could smoothly continue. What other reason would Barzini have to be interested in Tattaglia and Vito making peace? Tattaglia and Vito calling a truce makes it easier for Barzini to get what he wants.

Because Barzini was the only one insinuating don Vito was hording all the political power to himself, which is when don Vito let it be known that the connections were his and wouldn't be given to the families, but he made others confirm he had never denied an accommodation. He sensed a dislike for him and animosity from Barzini. And coupled with Barzinis string suggestions for the same of drugs along with the analogies of it being impossible not to engage in, led don Vito to have a eureka moment because he knew he was the on member that up until that point had been against narcotics. Barzini just campaigned for the drugs and access to the politicians too hard to the point it was obvious to don Vito since he knew the other guy wasn't that clever.

Two answers explained it very well, so without reiterating what they said, let me just add a quick tip about motivations and deductions. When trying to figure out an issue that involves people, one of the key things to be looking out for is who has the most to gain and who is the least and most likely suspects. In this case, as others stated, it appears that Barzini was in control of the meeting and would gain by having this negotiation completed. None of the other families were as positive and forthcoming about that point, so it is obvious he had something to gain. It is makes it even more suspicious that he would back this when he appears to have no stake or bias and would be considered the least likely candidate up to that point. Couple that with, well let's face it, they type cast the perfect guy as Tataglia because hell I think I could even take that guy down.... LOL!!