I think that to accuse Karl Marx of "antisemitism" is kind of unjust. Due to the great tragedy that supra-European antisemitism has wrought so recently in our times, it is perhaps hard for us to think in the mindset of the mid-nineteenth century. Today the very concept of a "Jewish condition" or a "Jewish question" would rightly be considered "antisemitic", yet these issues were very real and lacked contention to varying degrees in Central and Western Europe at this time. That it would be the Jewish question that would set the whole unholy engine in motion was neither to be known nor imagined.
The goal of the European nation-state was the equality of all within it before the law and the abolition of all special privileges held by members of the society. A class system developed which was defined mainly by the relation of each class to the other - the proletariat, the peasantry, the aristocracy, the bourgeoisie, and so on. But one group remained essentially un-assimilated and apart from the class system - very literally a "nation with a nation" - and these people were the Jews. Frequently proving themselves unwilling to engage in capitalistic enterprise and prohibited from developing an intelligentsia by the state (until the end of the 19th century), there was one property of the Jewish "nation within a nation" that the state found particularly useful: its supra-European nature.
As early as the 17th century, when flourishing and increasingly more expensive monarchial wars were draining countries dry, the Jewish state banker (often known as a "court Jew") was enlisted by the state to provide finance and to administer this finance. Thus a small group of Jews - apart from the rest of the impoverished European Jewry - began increasingly to be identified with the state. When the aristocracy lost its privileges, they became in fact the only group that was directly identified with the state. Thus in countries where attacks on the authority of the state were still considered profane, antisemitism was a replacement (in much the same way members of the nobility who were thought to influence the Monarchy used to be criticised by Medieval rebels, rather than the actual Monarchy itself)1.
So we may recognise two distinct components of Central and Western European Jewry in the time under consideration - the large mass of Jewry at various stages of emancipation, and the privileged few who by their services to the state already enjoyed emancipation, by which is meant equality with other citizens. The nation state's problem was that if emancipation of all of Jewry led to the death of the Jewish identity, the death of its financial networks would follow. The financial nets of the Jewish state bankers (of which the most famous, and infamous to antisemites, is no doubt the House of Rothschild) relied on the capital of middle-class Jews being drawn into the "net". The power of the Jewish state bankers rested on their position within Jewish society, and emancipation could only lead to a deterioration of their civic and social status. We see frequent examples of state officials not really protesting attacks on Jews in general, but moneyed Jews in particular.
Karl Marx's particular brand of writing on the Jewish question can be traced to a brand of anti-Jewish (note with care I do not say antisemitic) writing which was popular in Prussia shortly after the Congress of Vienna. Radicals (this sort of radicalism was concerned with rights for "the people" and "the nation") looking to attack the reactionary Prussian government, but to do so in an understated way, found it useful to focus on the Jews instead. They did not understand why it was that the Jewish people were preserved as a separate entity within the nation, but they knew they didn't like it. They recognised that the Jewish question wasn't just about human tolerance, but was entwined amidst the politics of the nation state.
Karl Marx's writings on the Jewish question separate it from the discussion of theology, and instead discuss the social aspect of European Jewry ("Let us consider the actual, worldly Jew ..."). He is discussing the social element of the Jewish condition which he believes prevents its emancipation, and unsurprisingly for Marx, his answer lies in the economics - namely, moneylending. Karl Marx was not a moneyed Jew, he was a Jewish intellectual, shunned by the state. His anti-Jewish writings were aimed at condemning the rich Jewish bankers, whom he and his fellows regarded bitterly due to their favour by a state which would happily see Marx and his ilk starve. Hannah Arendt writes that "Marx as an individual Jew was as little embarrassed by these arguments against 'Jewry' as, for instance, Nietzsche was by his arguments against Germany."
And that's why Karl Marx wasn't an "antisemite". He was merely representative of the Jewish intelligentsia which began to develop in the 19th century, which sought to differentiate itself from its maligned, richer fellows. Much of Marx's supposedly "antisemitic" tract in fact is a discussion of the Jewish relationship with a Christian state, and how true "freedom" for both Christians and Jews might be achieved.
1. The Jewish state bankers were utterly a-political as a group, being loyal rather to authority itself (as opposed to the masses, which they instinctively and rightly distrusted).
Sources
Arendt, Hannah. The Origins of Totalitarianism: Harvest, 1966.
Marx, Karl. On the Jewish Question: http://www.marxists.org/archive/marx/works/1844/jewish-question/ (1844)