In a general sense, Dickens created sympathetic characters mostly by making them vulnerable. Many of his most sympathetic characters are children.
Dickens used children because it is hard for us to blame them for their circumstances. If I was writing about poverty and used adults, people might dismiss them because they are adults, and just assume they are not working hard enough or are lazy.
Children, on the other hand, cannot fend for themselves. This is where Oliver Twist and David Copperfield come in. Tiny Tim had parents, but they were hard-working too. Bob Cratchit is partially a sympathetic character because he is a father. We sympathize for his children.
The best example of this is a quote from A Christmas Carol. Scrooge asks the Ghost of Christmas Present who the children belong to.
“They are Man's,” said the Spirit, looking down upon them. “And they cling to me, appealing from their fathers. This boy is Ignorance. This girl is Want. Beware them both, and all of their degree; but most of all beware this boy, for on his brow I see that written which is Doom, unless the writing be erased."(enotes etext p.40)
This demonstrates that Dickens felt we were all responsible for the poor, and especially the children.