In Claudius's soliloquy near the beginning of Scene III, what do we learn about Claudius's character?
I assume that you are talking about the speech in Act III, Scene 3. This is the one where Claudius is trying to pray, but cannot. This is the time in the play where I actually feel sorry for him.
What this speech tells us about Claudius is that he really has a conscience but that he is too weak or too greedy to actually do what that conscience tells him to do. He feels really guilty for having killed his brother and he wants to be forgiven. But he cannot really feel forgiven because he is not willing to give up what he has gotten as a result of the killing. He still wants to be king and he still wants Gertrude.
So here he is -- he knows he has done something really evil but he can't bring himself to give up the fruits of his crime. Pretty sad to be torn like that...
One other thing that we learn about Claudius that is interesting, particularly in the light of Hamlet's indecision and the way he compares himself to Claudius as a man of action, evil though it may be, is that Claudius too has problems making decisions. He states:
Pray can I not,(40)
Though inclination be as sharp as will;
My stronger guilt defeats my strong intent,
And, like a man to double business bound,
I stand in pause where I shall first begin,
And both neglect.
He too cannot always figure out exactly what to do as he neglects both sides of the decision. In the end of course he makes himself kneel to begin asking for forgiveness, but he is torn for a time about whether he can or ought to.