As the title says did he explain how it might be done and did he explain the image which would replace the conceptualisation of private property? In the Communist Manifesto he states that "The distinguishing feature of Communism is not the abolition of property generally, but the abolition of bourgeois property. But modern bourgeois private property is the final and most complete expression of the system of producing and appropriating products, that is based on class antagonisms, on the exploitation of the many by the few.
In this sense, the theory of the Communists may be summed up in the single sentence: Abolition of private property."
It is clearly important then, it is the essence of communist thought, but what exactly was envisaged in its place?
This section of Marx' Letter to Ruge is particularly illuminating so I'll quote it at length, but the whole letter is worth a read.
Not only has a state of general anarchy set in among the reformers, but everyone will have to admit to himself that he has no exact idea what the future ought to be. On the other hand, it is precisely the advantage of the new trend that we do not dogmatically anticipate the world, but only want to find the new world through criticism of the old one. Hitherto philosophers have had the solution of all riddles lying in their writing-desks, and the stupid, exoteric world had only to open its mouth for the roast pigeons of absolute knowledge to fly into it. Now philosophy has become mundane, and the most striking proof of this is that philosophical consciousness itself has been drawn into the torment of the struggle, not only externally but also internally. But, if constructing the future and settling everything for all times are not our affair, it is all the more clear what we have to accomplish at present: I am referring to ruthless criticism of all that exists, ruthless both in the sense of not being afraid of the results it arrives at and in the sense of being just as little afraid of conflict with the powers that be.
Therefore I am not in favour of raising any dogmatic banner. On the contrary, we must try to help the dogmatists to clarify their propositions for themselves. Thus, communism, in particular, is a dogmatic abstraction; in which connection, however, I am not thinking of some imaginary and possible communism, but actually existing communism as taught by Cabet, Dézamy, Weitling, etc. This communism is itself only a special expression of the humanistic principle, an expression which is still infected by its antithesis – the private system. Hence the abolition of private property and communism are by no means identical, and it is not accidental but inevitable that communism has seen other socialist doctrines – such as those of Fourier, Proudhon, etc. – arising to confront it because it is itself only a special, one-sided realisation of the socialist principle.
And the whole socialist principle in its turn is only one aspect that concerns the reality of the true human being. But we have to pay just as much attention to the other aspect, to the theoretical existence of man, and therefore to make religion, science, etc., the object of our criticism. In addition, we want to influence our contemporaries, particularly our German contemporaries. The question arises: how are we to set about it? There are two kinds of facts which are undeniable. In the first place religion, and next to it, politics, are the subjects which form the main interest of Germany today. We must take these, in whatever form they exist, as our point of departure, and not confront them with some ready-made system such as, for example, the Voyage en Icarie.
It's first of all important to understand what "private property" is for Marx. He explains this in the section on "The Trinity Formula" in Capital Volume 3:
However, capital is not a thing, but rather a definite social production relation, belonging to a definite historical formation of society, which is manifested in a thing and lends this thing a specific social character. Capital is not the sum of the material and produced means of production. Capital is rather the means of production transformed into capital, which in themselves are no more capital than gold or silver in itself is money. It is the means of production monopolised by a certain section of society, confronting living labour-power as products and working conditions rendered independent of this very labour-power, which are personified through this antithesis in capital. It is not merely the products of labourers turned into independent powers, products as rulers and buyers of their producers, but rather also the social forces and the future [? illegible] [A later collation with the manuscript showed that the text reads as follows: "die Gesellschaftlichen Kräfte und Zusammenhängende Form dieser Arbeit" (the social forces of their labour and socialised form of this labour). — Ed.] form of this labour, which confront the labourers as properties of their products. Here, then, we have a definite and, at first glance, very mystical, social form, of one of the factors in a historically produced social production process.
The bolded part is what is most important: Private property for Marx rests on exclusion. This means that whereever the means of production are monopolised by a section distinct from society itself, private property exists. It does not matter whether this instance is a person, a collective of some sort, a corporation or a state.
The question of private property is central, because it necessitates the producers exchanging their labour products in order to meet their needs. Marx argues that therefore production being carried out by isolated producers makes the products of labours "commodities", which means that additionally to their social utility, they acquire the property of posing the social relation "value". He explains that the simple equation of two products means that the respective producers abstract from the concrete labour that was necessary to produce the product, and that they therefore equate "abstract labour", labour-time in trade, which then becomes the substance of value.
Marx's work in "Capital" is not one of an economist. He is not trying to rival Adam Smith or David Ricardo, who he thoroughly studied, as much as all the other economists of his time. His endeavour is different:
The standpoint of the old materialism is civil society; the standpoint of the new is human society, or social humanity.
Taking on the "standpoint of social humanity" means to acknowledge that the way humanity currently lives is fundamentally inhuman, perverted, inverted. Marx's attempt is to explain what humanity has to do in order to escape from its self-created inhuman shell - he finds this inhuman, alienated way of living most adequately expressed in the categories of economics, as in productive, purposeful activity, humanity not only creates the world it inhabits, but also itself. His aim is therefore to show how the categories that economics expresses arise, and consequently how they're transitional, how humanity can do away with them in order to finally be at home in its self-created world.
The real abstraction "value", unconsciously brought about by the producers in trade, ultimately finding its most adequate form of expression in money, becomes the real subject that governs society. Even though humanity does create its world, its own powers confront it as an alien thing: the substance that is also subject; value in process: capital.