When Voldemort cast the Avada Kedavra and Harry didn't raise his wand in defence, Harry went into a kind of limbo where he met Dumbledore. When Harry awakens, he finds that Voldemort had also fainted. What had Voldemort experienced while he had fainted?
To answer this question, we can begin with the book reference to the scene:
The Death Eaters had been huddled around Voldemort, who seemed to have fallen to the ground. Something had happened when he had hit Harry with the Killing Curse. Had Voldemort too collapsed? It seemed like it. And both of them had fallen briefly unconscious and both of them had now returned.... (Deathly Hallows - The Flaw in the Plan)
When Voldemort "killed" Harry, he destroyed the Horcrux, but Voldemort should not have felt any pain due to this event because he cannot feel when the Horcruxes were destroyed:
But surely if the boy had destroyed any of his Horcruxes, he, Lord Voldemort would have known, would have felt it? ... How could Lord Voldemort not have known if he, himself, most important and precious, had been attacked, mutilated? (Deathly Hallows - The Final Hiding Place)
So I think it is safe to assume that, at least physically, other than the actual fall, he experienced little to no pain.
As for what happened to his consciousness during the time that Harry was in King's Cross (Limbo), we can take a look at what J.K. Rowling has said on her old website:
...It is the last piece of soul Voldemort possesses. When Voldemort attacks Harry, they both fall temporarily unconscious, and both their souls – Harry’s undamaged and healthy, Voldemort’s stunted and maimed – appear in the limbo where Harry meets Dumbledore.
So, Voldemort also appears in King's Cross along with Harry while his physical body is left unconscious in the Forbidden Forest. Harry's entire soul is present in limbo, as well as the remaining bit of Voldemort's.
The final question to answer is was Voldemort's mind present in the soul in limbo? In other words, was he aware of the scene happening or the conversation between Dumbledore and Harry?
I would argue no, he was not aware, because of the state of his soul that shows up at King's Cross:
It had the form of a small, naked child, curled on the ground, its skin raw and rough, flayed-looking, and it lay shuddering under a seat where it had been left, unwanted, stuffed out of sight, struggling for breath. (Deathly Hallows - King's Cross)
The manifestation of Voldemort's soul was very broken, hidden, and practically dying. As Richard pointed out in the comments to this answer, it seems very unlikely that this manifestation of Voldemort's soul was capable of eavesdropping through its suffering. In Dumbledore's words (in the same chapter) it is:
Something that is beyond either of our help.
Voldemort, when he is awoken, also does not seem to give any indication of that scene. If he had been present and witnessing the conversation between Dumbledore and Harry, it is likely he would have been much more skeptical of Harry's death.
The final bit of information that could make the case that Voldemort did in fact know what happened is:
... small comfort though it was, that Voldemort was wary of approaching him, that Voldemort suspected that all had not gone to plan....
It seems to me that Voldemort only suspects things might not have worked completely according to plan because of his past problems trying to kill Harry, not because he witnessed the scene in King's Cross.