How prominent? Well, Karl Marx was prominent in many ways, but in all but one, his radicalism followed him, and was part of why he was prominent.
First, there was the young Marx. A relatively unknown young student, who started making a name for himself in some radical circles, most notably the "doctor's club" known as the 'Young Hegelians' around Bruno Bauer and Ludwig Feuerbach, both of which would have a large influence on Marx, but who he also later distanced himself considerably from. Marx's doctoral thesis also garnered some controversy, particularly among conservatives, due to its... not positive view of religion. So Marx became known, while still fairly young, in academic circles. But not enough to be prominent, perhaps more of a name that some would be able to recognise.
Then Dr. Marx started doing journalism. Radical journalism. As he returned to his native Rhine, he started writing for the creatively named Rheinische Zeitung(Lit. "The Rhenish Times,") where he clearly started advocating for socialism, and opposition to not just the Prussian state of affairs, but the bourgeois state of affairs as a whole. His name was spreading in radical circles, and not just on the lips of academics any longer. Most unfortunately however, his name reached the ears of the Prussian commissioners and censors, and unlike his earlier student pranks, this time, the authorities likely wouldn't be satisfied with throwing the young delinquent into a cell for the night. All issues were put under harsh scrutiny, which Marx lamented, as the ardent supporter of freedom of speech he was his entire life, and when the RZ published an article criticising the Tsar of Russia, they fate was sealed. The Tsar requested the radical paper shut down, and the Prussian state happily complied. But now Marx was a known writer, particularly in the Rhineland, but also starting to be known in radical circles, outside of Germany.
Marx and his family then went into exile, to Paris, the hub of European radicalism. Here Marx was co-editor of the Deutsch-Französische Jahrbücher("German-French Yearbook"), a radical journal, which, despite only being published in one issue, was relatively successful. After the DFJ, Marx went on to the German language radical newspaper Vorwärts("Forwards,") which is where he also really starts with one of the defining parts of Marx's writings: the critique of not only reactionaries and bourgeois politics, but also progressive liberal and utopian socialists came under Dr. Marx's scrutiny. His name grew in recognition, and he was a regular not only the salons of the radical academia, but also among workers directly more often now, attending meetings in various workers associations. But as fate would have it, another foreign monarch, this time the King of Prussia, requested that Marx's newspaper be shut down, and Marx was expelled from Paris, and him and his family went into exile in Brussels.
From Brussels, Marx started to write some of his more famous works, like The German Ideology and The Poverty of Philosophy, the first outlining some key Marxist ideas, and the latter criticising Proudhon's petty-bourgeois philosophy, anarchism or mutualism or whatever terms we prefer. The former was not published until 90 years later, almost, but the latter was published, and was read widely, but never republished in Marx's lifetime. From Brussels, Marx was famous enough to be invited to England, to meet with leaders of the Chartist movement, and with his new friend who he met in Paris, Engels as guide and translator, the trip was a success. It was also here that the Communist League, Marx and Engel's party, was organised on the basis of the secret society called "The League of the Just," started to grow, and they wrote the famous Manifesto of the Communist Party, which was, over the next years, translated into multiple languages. During the revolutions of 1848, Marx was accused of arming Belgian workers, using part of his inheritance, expelled from Brussels, and he returned to his dear Paris, but quickly left again for his native Rhine, where revolution was also brewing.
In Cologne, Marx was known already. He set up a newspaper, the Neue Rheinische Zeitung("New Rhenish Times,") a creative name, meant only to be defiant, and his paper had, during the Revolution, thousands of daily readers. Marx again ran head first into the Prussian authorities, and regularly found himself in court, needing to defend his radical writings and his paper, which also contributed to his fame as a revolutionary. But as the reactionaries won out in the revolution, Marx was forced to flee the country, back to Paris, which was also under the boot of reaction (and cholera), so from there, onto London: the Marx family's final home.
In London, Marx was already a known name. The Manifesto had been published in German, Danish and Polish—foreigners made, and make, up a considerable portion of London's working class—in London, and would soon also be published in English as well. He was known among radicals, big and small. It wasn't a hero's arrival though, don't get me wrong. But Marx had international fame at this point. And from 1852, Marx started writing for the American newspaper, the New-York Daily Tribune, which spread his name across the Atlantic. Though here it wasn't as a radical he became known, but mostly as a reporter on European affairs, and well liked by the paper's readers. Not to mention his writings in plenty of journals and newspapers in a lot of other countries, from the US to Germany, to even some journals in Russia, in places with strict censors under pseudonym. In London many of his "older" works were written, most famously Capital, the first volume even having multiple issues within Marx's own lifetime. Marx—and Engels—also corresponded with leaders of the young socialist parties across Europe, and for a while, the young Social Democratic Party of Germany, was rumoured to be led by "the two old men in London." Marx was, by now, in the last 20 years of his life, an international name, known across Europe and North America, censored by states like Russia, though they did allow the publication of Capital, and a respected name in the international socialist community, despite the collapse of the 1st Internationale, due to differences between Marxists and Bakunists. Marx was prominent, and he knew it, and, I think, he liked it. As the story goes, his last words would be that "last words are for fools who haven’t said enough," so, should this be true, he obviously didn't see himself as done... which the huge amount of unfinished work, and notes to projects he hadn't even started, and letters not send, he left behind also proves.
Sources:
Hobsbawm, Eric. How to Change the World: Tales of Marx and Marxism. 2011. Little, Brown.
Liedman, Sven-Eric. A World to Win. 2018. Verso.
Sperber, Jonathan. Karl Marx: A Nineteenth-Century Life. 2013 W.W. Norton & Co.
The MEGA, and they continue to publish, contains a lot of Marx's journalism and his letters.