Until this scene, Hamlet is clearly the underdog because he does not know with absolute certainty that any of what he is acting on is true. Claudius has the upper hand because he is perfectly aware not only of his own actions but of the probable import of Hamlet's actions. (In other words, he's pretty sure Hamlet knows the truth, or at least suspects it.)
After this scene, the playing field seems about even. Hamlet is sure about what Claudius has done, and Claudius knows Hamlet is finally ready to act on his suspicions. What ensues, then, is a game of cat and mouse in which each is trying to outmaneuver the other but neither has a true advantage.
Hamlet chooses not to kill Claudius at this point because he doesn't want to kill Claudius after Claudius has purged his soul in prayer. He wants Claudius to die in a state of sin as King Hamlet did. He tells us this when he says, "To take him in the purging of his soul,/When he is fit and seasoned for passage?/No."
In reading the above posts, I thought of something that never occurred to me before. Hamlet has the best chance to kill Claudius here, but he decides not to, in faith that he will have a better opportunity to make a better kill. In this sense, he almost could be judged as "cocky." This is the KING OF DENMARK that he's got to kill. You would think that he would have to take his chances when he could. If you think of Hamlet as "cocky" then, could you say that Claudius is then the underdog? lmetcalf's comment that "Hamlet seems like the underdog when you consider all YOU know, versus all that HAMLET knows" is key to answering this question.
You could also consider dramatic irony and see that Hamlet is the underdog. We know that Claudius is plotting behind Hamlet's back and seems to have a political justification for his actions. Rosencrantz talks about how a King is important to the whole kingdom and "the cease of majesty dies not alone, but like gulf doth draw what's near with it." He uses that metaphor to say that kings have to do what they must because their failures (death) results in catastrophe for all the people of the kingdom. With that kind of logic, Hamlet doesn't stand a chance that Ros and Guil will be loyal to him -- they have clearly made their allegiance to the King known.
Once Claudius is alone and "praying" we know that he can't sincerely pray because he isn't willing to give up what he has gained for his crimes, but Hamlet doesn't know the insincerity of the prayer and chooses not to kill Claudius when he has this easy opportunity. This choice ends up leading to the downfall of Hamlet -- he goes on to accidentally kill Polonius, and allows Claudius more time to actively plot his death. Hamlet seems like the underdog when you consider all YOU know, versus all that HAMLET knows.
I guess this depends on what you mean by "underdog." Just in this scene, Hamlet clearly has the power over Claudius, at least for one moment. Hamlet could have killed Claudius while Claudius was praying. Since he had the power of life or death over Claudius at that point, you could certainly argue that Claudius was less powerful than Hamlet. If that makes him the "underdog" then I would say that Claudius is the underdog in this scene.