I think I would say that he feels kind of conflicted about the whole thing (or at least he has been in his life).
He does not answer when people ask him the question, but he has been dealing with it all his life.
To me, the answer he really gives is his idea of the double self that he says black people in America have to deal with. He says that he has to work hard to try to bring together the two parts of himself that are kind of separate -- the part that is black and the part that is American.
So, to me, this is his answer to the question of what it feels like to be a problem. It is a hard thing to deal with because he has to feel like there are two parts of him that are both him, but are not always compatible with each other. He must find a way to merge the two.
Not sure I understand your question or agree that DuBois was a problem - he was certainly an activist, and a thorn in the side of Booker T. Washington and the segregation establishment, but I wouldn't call him a problem.
As an activist, he prided himself for his principled stands for equality. He felt that all American citizens should be treated equally, and that if the government would not give them legal equality in the courts, then they would find equality in the economic mainstream. DuBois was a socialist for this purpose - he felt that once blacks had economic equality, then social and political equality would follow. This is essentially what did happen in the 1950's, as the effect of boycotts by civil rights groups had enough bite to change some stubborn southern minds. Of course, by that time, DuBois was long gone, having moved to Ghana as a permanent expatriate where he founded the Ghanaian Communist Party.