[Please read "Libertaria: A Libertarian Paradise"]
Libertarianism is capitalism with minimal government. The focus of libertarianism is on why there should be only minimal government and what it should be limited to doing (protecting the nation from foreign attack and enforcing contractual agreements.)
There is nothing in the philosophy of libertarianism that challenges capitalism or that challenges the class inequality that is inherent in capitalism.
Libertarians emphasize individual freedom, meaning freedom from government coercion and freedom from aggressive violence from any other person. They emphasize the importance of people having no obligations except those agreements made with others freely and by mutual agreement, voluntarily. So how does class inequality arise from this?
Class inequality arises from this libertarian kind of society because a) there is no limit on how much property (including the means of production) an individual (or group of individuals) may own, and b) this libertarian society is based on buying and selling things (with money) in a free market. If one imagines such a libertarian society in which, initially, all individuals are equal with respect to the amount of wealth they own, it is not hard to see how a capitalist class and a working class will inevitably emerge. Here is why.
Some people are lucky and others unlucky. One person's house burns down and another person's doesn't. One person has good crops and another one has crop failures. What happens? The unlucky person, quite voluntarily, must make a deal with a lucky person to get out of a hard spot. The lucky person is in the driver's seat, able to take advantage of the hardship of the unlucky person. The result is that the lucky person comes out the winner, the unlucky person the loser; the lucky person gets wealthier and the unlucky person poorer. Wealth inequality inevitably emerges because there is nothing in the libertarian morality that prevents it from emerging.
Likewise, some people have many siblings and others none. The "only child" inherits more than the child with many siblings. More wealth inequality develops.
By countless means, some become wealthier than others. The wealthier ones thereby have more power and can take advantage of more opportunities. Eventually they buy more means of production, while the unlucky ones are forced to sell their means of production (land, tools, etc.) to make do in hard times. Wealth begets more wealth. The rich get richer, the poor get poorer. Nothing in libertarianism prevents this. Full-blown class inequality inevitably emerges. Libertarians do not consider this a problem. As long as the government is minimal and nobody uses violence to coerce anybody and all contractual agreements (enforced by the government with violence when necessary) are entered into voluntarily, then all is well according to libertarians. But class inequality is not ok according to egalitarians.
Click here for an article about how class inequality develops inevitably in a pure libertarian society.