In Animal Farm, how does Orwell demonstrate Marx's claim, “The history of all hitherto existing society is the history of class struggles"?
We can look at the class struggle Marx was talking about during his time as the bourgeoisie (the owning capitalist class) and the proletariat (the working class). The bourgeoisie controlled the means of production, hoarded all of the surplus profit, and the proletariat made just enough to survive. In the novel, Farmer Jones (and other farmers) are the bourgeoisie and the animals are the proletariat, the working class.
In The Communist Manifesto, Marx foresaw the proletariat taking over the means of production when the bourgeoisie became unfit to rule. Marx believed this class struggle (between the Proletariat and the Bourgeoisie) was similar to previous class struggles in history but that such a revolution by the workers would eventually result in a continual process that would lead to Socialism and then Communism: a progressive continuum resulting in better conditions for the working class (for the animals in the novel).
In Animal Farm, this is the original goal of the Animal Rebellion, but Napoleon turns the rebellion into his own dictatorship: comparable to how Stalin corrupted the ideals of the Russian Revolution. In both cases, with Russia and in the novel, the workers (the masses) were left with a situation just as bad as they had been under the previous rulers (for Russia with the czar, for the animals under Farmer Jones).
Under Farmer Jones, the animal class struggled against his (owning) class. Once Napoleon took over complete control, the animals were in a similarly bad position struggling against his oppressive regime. Orwell was not favoring one form of government over another (capitalism or socialism or communism). He was criticizing the Russian Revolution for abandoning its ideals: taking care of the workers, the majority of the people. This progression in the novel of oppressor vs. oppressed (Jones vs. the Animals and Napoleon vs. the Animals) represents Marx's idea that class struggle has continued. Marx foresaw an eventual (down the road) classless society; this is why he used "hitherto" or "up until now" to describe the history of class struggle. He thought the revolutions leading up to communism would result in a more equal, classless society; such was the original goal of the Animal Rebellion. Orwell criticized Russia via novel for failing to continue towards a classless society. Instead, Stalin (and Napoleon) replaced an oppressor with another oppressor, thus continuing the cycle of class struggles, thus continuing the struggle between oppressor and oppressed.
In the last chapter, Mr. Pilkington confirms the newest class struggle by acknowledging to Napoleon:
If you have your lower classes to contend with . . . we have our lower classes.
This is previously culminated in condensing the Seven Commandments to one: "ALL ANIMALS ARE EQUAL BUT SOME ANIMALS ARE MORE EQUAL THAN OTHERS." This clearly shows that another division of classes has formed and therefore another class struggle.