How did nineteenth-century advocates of Marxism, conservatism, and liberalism view human nature and define the purpose of the state?

First, one should note that "liberal" and "conservative" positions meant something quite different in the 19th century than they now mean.

Liberals who were members of the Whigparty tended to be drawn from the middle classes and the wealthy manufacturing families rather than either the poor or the nobility. They tended to believe in individual freedom and in reducing the role of the state. They were strong advocates of freedom of religion. They also...

First, one should note that "liberal" and "conservative" positions meant something quite different in the 19th century than they now mean.

Liberals who were members of the Whig party tended to be drawn from the middle classes and the wealthy manufacturing families rather than either the poor or the nobility. They tended to believe in individual freedom and in reducing the role of the state. They were strong advocates of freedom of religion. They also opposed any laws regulating commerce or labor. They had a generally optimistic view of human nature, often expressing itself in a belief in the natural tendency of markets to create wealth (Adam Smith's "invisible hand").

Conservatives or Tories tended to distrust human nature and see the need for a strong state to protect the poor and restrain the rich and powerful. They supported the notion of an established church in England and were less concerned about freedom than about justice and order. 

Marxists saw the state in its nineteenth-century form as an instrument of bourgeois oppression of the masses. Although Marxists believed in a strong centralized state, they thought it should be controlled by and benefit the laboring classes rather than the rich. Marxists saw class and economic situation as in many ways determining human nature.