Oh poor, sad Jo. He can't even afford an "e" at the end of his name. He is so pathetic and ridiculously guilt-inducing that it's hard to discuss him as a character without straight out crying. But here goes.
The lowest of the low, Jo is an illiterate, destitute, orphaned boy who sweeps the street for handouts. He is totally vulnerable and unprotected from people, disease, and the horrors of poverty. Even his key knowledge of some of the facts that unravel the novel's mystery only brings him more trouble.
In all of his fiction and much of his nonfiction, Dickens was a very outspoken advocate for the poor – especially poor children. It's hard to imagine now, but he lived at a time when there were no systems of intervention to deal with children who were orphaned or abandoned by their parents. No options – just being in the street and struggling to get by. That's what happened to Jo, who is forced to rely on the handouts of strangers for survival and is constantly being harassed by the constables (police), who have the authority to make him move from one spot to the next for loitering.
At the same time, Jo might well be the most important character to the novel's plot. His existence serves to connect people who otherwise couldn't and wouldn't ever meet. At the inquest, he links Nemo, Tulkinghorn, and Snagsby. By showing the veiled woman around, he links Lady Dedlock with Snagsby and Guster. Snagsby's charity to Jo links Guppy with Snagsby and the Chadbands, which in turn leads Guppy to figure out the Esther and Lady Dedlock relationship. By bringing medicine to Jenny, Jo is linked to the neighborhood around Bleak House, and so to Esther and Jarndyce. And so on and so forth. So this insignificant and powerless boy is essential to the novel's most important events.
At the same time, Jo is totally removed from all the things that create society and bring people together. He hasn't had any religious education, and no Christianity means no fellowship or belonging in Victorian England. Not just that, but no Christianity also means he can't participate in normal civic life – for instance, he can't testify at the inquest of Nemo, even though he is the only person who knew him. The magistrate simply doesn't believe that a boy like him, who hasn't learned to fear hell, can be relied on to tell the truth. Tulkinghorn also exploits the fact that Jo is a non-person in the eyes of the law to get information out of him, then shove him out the way when it's convenient.
Jo's death scene is a classic. It's a masterly combination of a realistic, poignant, and incredibly melodramatic end to a wasted life marked by genuine goodness. It's a piece of propaganda made to yank on readers' heart strings mercilessly. How does this work? Why, despite being told that Jo is dirty, ugly, and covered with sores, aren't we repulsed by him? What do you make of the fact that Jo doesn't get to finish saying the prayer that Woodcourt is teaching him?