Marx talks more about fiat currency in the (unfinished) Foundations of the Critique of Political Economy, which is more or less his notes for writing Capital later on.
Fiat currency is money backed by nothing, except a decree by the local government that the bills have value. This is the system the world uses today (known as the Bretton-Woods system).
Back in Marx's time, most nations, and especially the major ones, used representative currency, which is money that represents a government's physical gold reserves (or other similar commodities, like silver, but usually the main commodity was gold).
Theoretically, in a representative currency system, you can only print as much money as your reserves. If you want to print more, you need to either go into debt so you can buy more gold, or you will need to debase the value of your currency (i.e. say that 1 pound = 1 gram of gold today, but from tomorrow onwards 10 pounds = 1 gram of gold). In a fiat currency, you don't have to worry about the value of your gold reserves, so you can just print money when it is needed. For example, if you want to stimulate the economy.
In Marx's time, the economists supporting the fiat currency system would argue that since people still mined for gold, the amount of gold was constantly increasing, and its price was decreasing. This created a naturally increasing inflation that was hard to control. With a fiat currency you wouldn't have to worry about that at all, since the government could control inflation by just deciding not to print more money (at least that was the general idea). And it would make the economy more robust and able to deal with quick crises demanding quick influx of cash, like a crop failure. There were also some arguments about easier trade, etc.
As usual, Marx correctly predicts that, because fiat currency is backed by nothing except a government's decree that it is legal tender, then there is no limitation on how much currency can actually be printed/minted. In fact, this is precisely the reason for which a government creates a fiat currency. Hence, as governments won't be able to resist the urge to print in a nigh unlimited fashion, inflation will rise perpetually, causing an equally perpetual increase on the cost of food, products, housing, clothing, etc, as well as the source of resources, machinery, labour, etc.
This will affect both the proletariat and the bourgeois, but in different ways.
The proletariat will see the value of their wages diminish every year, and will face an increasing need to enter into debt in order to survive. On the other hand, the bourgeois will see the value of their investments and deposits also diminish every year, and will demand constantly an expansion of the economy, so as to minimize the effects of inflation.
For a nation without colonies, dependencies, protectorates, and spheres of influence, expansion of the economy means national indebtedness to other bigger powers, so as to have the funds to actually expand the economy. For a major power, expansion of the economy means using its military and diplomatic leverage to expand its influence, invade, colonize, so as to open for itself more markets, and acquire cheaper resources.
For all nations, expanding the economy means finding ways to squeeze even more profit from the productive processes, which means putting more pressure on the proletariat, forcing them to work harder, while compensating them for even less.
Simultaneously, inflation will negatively impact banks the most, as banks rely the most on the value of their currency reserves (which is diminishing due to inflation), and on the value of their assets, i.e. debts (which is similarly diminishing as inflation rises). Hence, banks will enter a perpetual cycle, where their lending fees (interest rates) become ever-more increasing, while their clients' interest rates on deposits will become ever-more decreasing. This will lead into an even bigger increase in the price of products, compounding the same problem perpetually.
Therefore, eventually, a nation employing a fiat currency scheme discovers that, even though in theory the fiat currency is backed by nothing than the government's word, in reality the fiat currency is backed by the entire nation's economic output. Any efforts to limit inflation, including limiting the previously unlimited printing of money, will wreak havoc on the economy. Any efforts to return to a representative currency (e.g. backed by gold) will cause the economy to contract massively, which is against the bourgeois' interests.
Whatever, the outcome, the result is the same: The economy's biggest commodity becomes debt. Debt owed by the proletariat to the bourgeois, nationally. Debt owed by minor nations to major powers, internationally. Thus, the bourgeois is further alienated from the product of its labour, while nations are alienated from their own economies.
This is not to say that a representative currency system won't cause the same effects. According to Marx it does. But a fiat currency would skyrocket debt, causing even more severe effects.
This is analyzed further by Lenin in Imperialism: The Highest Stage of Capitalism.