I have studied Marx in a lot of subjects in the school, such as History, Sociology, Economy and Political Sciences. The bad side of this is that each professor told me different things about Marx.
There are many Marxisms - philosophical, cultural, political & economical. Its important to see that Marxs own writings are part of what is now called political economy; for example his labour theory of value is an extension of Adam Smiths theory; and as a young man he was part of that group called the Young or Left Hegelians that was critical of the Prussian state and of Religion; its important to realise that as a thinker he drew on many strands of intellectual thought and because he was a point of reference for many other thinkers - some of them far from his own - he left his mark on many others.
Was he a defendor of capitalism?
When Marx was writing the system that is called Capitalism was prevelant in a small number of North European nation states, predominantly England; though he traced it back to the commercial city states of Northern Italy; this was one of the reasons he emigrated to England - to study its effects and causes at first hand.
Capitalism should be distinguished from the petty capitalisms of the classical civilisations of Islam & Rome; the immediate precursors of Capitalism. Its also worth noting that contra the physical & biological sciences that his science isn't attempting to uncover the laws of motion for all economies for all time - it is not a universal theory as Newtons theory of Gravitation was a universal theory - but that it was a historically & socially constituted description & critique; he wrote that a new economic & political order would demand a new analysis.
As an astute observer of the Capitalism regime he predicted that it would cover the globe - ie what is now known as Globalisation (in at least some senses). He also admired the energy of the bourgeois (as opposed to the feudal order) which he considered would transform the relations of production throughout the globe; part of this must be at least predicated on the global extent of the colonial empire which would ease the introduction of these new methods.
[was he] just criticizing the distribution of the income?
In the latter half of Capital (volume 1) he describes his anger at the inequity of Early Capitalism; so, yes; there is a moral imperative there; but this moral anger was shared by a wide spectrum of people in 19C England/Europe - for example Chartism;but its not simply about distribution of income;
and set a new political-economic system (communism)?
Marx was a marginal figure in the political scene in Europe - "he had been exiled from Germany, thrown out of Belgium and expelled from France" when he joined the First International - The International Workingmans Association; it was through this organisation that his ideas became visible.
Here Marx became a political activist and he aligned himself with the political current that became Communism.
Was the socialism installed in the Soviet Union by Lenin what Marx would like?
Its difficult to judge from Marxs own writing as he died well before the Russian Revolution. One suspects he would have celebrated the revolution but been dismayed by its trajectory - particularly during the Stalinist era.
Is right to say that Marx wasn't a communist?
Yes & No; as a political activist he was - he & Engels cowrote The Communist Manifesto after all; As a thinker and philosopher - if one is to take his theory as granted, and name the political & economic order that supersedes Capitalism as Communism then he could not be - its impossible to belong to a movement or order that is not in existence; what one can say is that he aligned himself with forces that he believed in bringing a new order about.