How did Dickens use the Cratchit family to the show the struggles of the poor?

In A Christmas Carol, Charles Dickens illustrates the struggles of the poor through his characterizations and descriptions of the Cratchit family.
The subject of poverty first appears at the very beginning of the novel when Scrooge is approached by two gentlemen taking up a collection to buy provisions for the poor over the holiday. When asked to help the less fortunate, Scrooge replies, “Are there no prisons?... And the Union workhouses?... Are they still in operation?” These questions suggest that poverty is the fault of those afflicted by it and that they somehow deserve their suffering. During the Victorian era, when A Christmas Carol was written, this was a common assumption.
But Dickens refutes this idea by his portrayal of the Cratchits. We first meet Bob Cratchit in Scrooge’s counting house, working diligently in a dismal setting despite the cold and the fact that his employer gives him only a tiny fire to keep warm. When we see his home in the third stave, we see it is a place of similar deprivation. Bob Cratchits is described as wearing “threadbare clothes darned up and brushed, to look seasonable,” while his wife is “dressed out but poorly in a twice-turned gown, but brave in ribbons, which are cheap and make a goodly show for sixpence.” The children’s garments are no better: Belinda is “also brave in ribbons,” and Peter is wearing his father’s shirt collar to dress up for Christmas. Dickens writes,

They were not a handsome family; they were not well dressed; their shoes were far from being water-proof; their clothes were scanty; and Peter might have known, and very likely did, the inside of a pawnbroker’s. But, they were happy, grateful, pleased with one another, and contented with the time...

These dire straits are expected, considering that Scrooge pays Cratchit a ridiculously low wage. But despite their poverty, they do their best to maintain appearances and remain cheerful. Their holiday meal is modest, but they do not complain. “It would have been flat heresy to do so.”
Their situation is not the result of a lack of industry. In addition to Bob’s toil at Scrooge’s counting house, Martha contributes from her poor pay as an apprentice milliner. Peter takes items to the pawn shop to help make ends meet, but still they cannot afford proper medical care for Tiny Tim, and their poverty will be the cause of his death. When Scrooge asks the Spirit of Christmas Present if Tiny Tim will live, the ghost's reply is grim:

“I see a vacant seat,” replied the Ghost, “in the poor chimney-corner, and a crutch without an owner, carefully preserved. If these shadows remain unaltered by the Future, the child will die.”

All the hard work in the world could not lift a family out of poverty if employers like Scrooge did not pay a fair wage. In Victorian-Era London, the poor struggled against their financial circumstances, the prejudicial assumptions of society, and a system that punished poverty with prisons and workhouses.

In the classic story A Christmas Carol, Charles Dickens depicts the Cratchit family as hard-working, industrious people who cannot escape poverty and are victims of unscrupulous, tight-fisted businessmen like Ebenezer Scrooge. Despite the fact that Bob Cratchit is a loyal, active employee and works over sixty hours per week, he struggles to put food on the table. Bob's daughter, Martha, is also portrayed as an industrious young lady who works as an apprentice at a milliner's while her brother, Peter, plans on becoming employed and contributing to his family's finances. The Cratchit family is also comprised of honest, likable people, who cannot escape poverty.
Dickens was very much aware of his upper-class Victorian audience and used the Cratchit family to gain their sympathy for industrious lower-class citizens living in poverty. In A Christmas Carol, Dickens challenges the widespread prejudiced belief that citizens were poor simply because they were lazy by characterizing the Cratchits as hard-working people. Dickens's goal was to promote social change by appealing to the upper-class members of society and influencing them to sympathize with lower-class citizens. Through his stories, Dickens continually portrays poverty-stricken, lower-class characters as likable, determined individuals who are the unfortunate victims of the economy and unscrupulous businessmen.

Dickens uses the Cratchit family to make the general point that even hard-working, decent people can easily fall victim to the scourge of poverty. Many of Dickens's readers will have shared the widely-held prejudice that poverty was the result of moral weakness or laziness. That's certainly an opinion shared by Scrooge himself. But in the case of the Cratchit family we can see that this is, at best, a gross over-simplification. Bob Cratchit works every hour God sends and yet his family are still dirt poor. This isn't because Bob's lazy, but because Scrooge is too tight-fisted to pay him a decent wage.
In presenting Bob Cratchit as such a decent, hard-working family man, Dickens wants to challenge his readers' prejudices; to get them to see that doing the right thing by society, by being thrifty and industrious, isn't always enough to save a family from poverty.