Du Bois begins the “forethought” of The Souls of Black Folk by declaring that “the problem of the Twentieth Century is the problem of the color-line." In his first chapter, he shifts the word “problem” so that it applies to himself; he describes how whites will often ask him some variation of the question, “How does it feel to be a problem?” Describe what Du Bois means by each instance of “problem,” being careful to note the difference between the two uses of the word.

In the first mention, the "color-line" is, in context, framed as a social "problem," something to be solved through policy and mutual understanding. In context, DuBois published The Souls of Black Folk at the height of the Progressive Era, when educated men and women turned their talents to come up with institutional solutions to the pressing issues that confronted society. These included such things as alcohol abuse, poor working conditions, crowded inner cities, and many others. Writing in 1903, at the beginning of a new century, DuBois was saying that race relations were the "problem" of the present and the future, and The Souls of Black Folk was an attempt to help people understand the nature of the problem by helping readers learn about African-American identity.
In the first chapter, he uses the word "problem" differently. The question often posed by well-meaning whites is, "what does it feel like to be a problem?" In this sense, DuBois and African-American people are "problems" in the sense that they have, as DuBois writes, a "double-consciousness." They live behind a "veil" that separates them from the world of white Americans. They feel a "twoness...an American, a Negro; two souls, two thoughts, two unreconciled strivings; two warring ideals in one dark body, whose dogged strength alone keeps it from being torn asunder." In this chapter, DuBois uses "problem" in this sense—a unreconciled search for "true self-consciousness" that exists in the minds of African-American men and women.
https://www.gutenberg.org/files/408/408-h/408-h.htm

Du Bois' use of the term "problem" highlights what he sees as the unique duality of Black subjectivity in America. The "problem of the color line," as he puts it in his Forethought section, alludes to a particular set of socioeconomic and political issues that stem from the history of Black people in America, particularly the legacy of slavery and the failure of emacipation to address basic inequities. When Du Bois shifts, in the first chapter, to thinking of himself as a "problem," he is describing the sense of alienation Black people feel personally as a result of that history. Du Bois feels, even in grade school, that others see him not as an individual, but as part of the larger "problem" of race. As a result, he comes to evaluate his own individuality in terms of his relationship to that problem. As Du Bois puts it, "one ever feels his two-ness,—an American, a Negro; two souls, two thoughts, two unreconciled strivings; two warring ideals in one dark body, whose dogged strength alone keeps it from being torn asunder."

There is a difference between the personal sense or perception of being a "problem," as expressed in Chapter 1: Of Our Spiritual Strivings, and the "problem of the color-line," as mentioned in Chapter 2: Of the Dawn of Freedom. However, the former condition was precipitated by the latter. DuBois's sense of being a "problem"—a notion that is imposed on him by the liberal, well-meaning whites he alludes to in his prose—is a result of the history that he describes in the second chapter, which constructed black people, first, as non-citizens and "three-fifths" of a person, then, a system that imposed a second-class status and eliminated due process.
Being unwanted, being a "problem," can result in a feeling of defeat or, in the case of DuBois, in a determination to strive: "That sky was bluest when I could beat my mates at examination-time, or beat them at a foot-race, or even beat their stringy heads." Still, the historical framework that constructed him as a "problem" causes that striving to be for naught in many instances: "Alas, with the years all this fine contempt began to fade; for the worlds I longed for, and all their dazzling opportunities, were theirs, not mine."
The perception of DuBois and other black people as a "problem" prevents whites, based on his anecdotes, from speaking to black people as individuals. Instead, black people are spoken to within the context of their racial oppression, from the understanding of their position as members of a permanent underclass. Thus, the "problem" of one's existence as a black person is created by the conditions of history and reinforced even by those who mean well.
https://www.bartleby.com/114/index.html