I think that it either started with Plekhanov, who was a fanatical Westernizer in Russia, or with Lenin himself. The practice is both fanatical and it does a disservice to Marx and his thought. Treating someone like a god is a negative complement. And people at the time must have known that there was something wrong with treating an author like that; human nature hasn't changed that much.
The other candidates, kind of far behind Lenin and Plekhanov, are people associated with the Social Democratic Party in Germany, particularly Karl Kautsky. But there's a difference between the Social Democrats of Germany and the Leninist Bolshevik party in Russia, which is typical of all European socialism, and that is that they didn't consider the movement to live and die based on Marxism. Socialism was a preexisting thing that Marx contributed to but which he didn't decisively define. The socialist movement could go on without doctrinaire Marxism, which it has albeit in a currently extremely watered down form. On top of that, the parts of Marxism that seem to be taken as official doctrine in the Social Democratic Party of Germany weren't the same ones that Lenin took to be essential; instead of Lenin's reading they were truer to the current centrist to conservative reading of Marx present on the Continent. So it isn't like there was international agreement on what exactly Marx meant when people quoted him chapter and verse.
I think it's better to anchor ones self in an understanding of the socialist movement, which included Marxist elements, as well as mainstream social democratic ones, Anarchist ones, ones based primarily on trade unions, and ones that were a sort of creative synthesis of a few of these, than it is to be totally Marx centric if you like the thought of Karl Marx.