Almost immediately. Alexander did not leave any heirs and following his death his generals split the empire among them. Some of those empires lasted a fairly long time though (for example the Ptolemic kingdom in Egypt lasted until the Romans conquered almost 300 years after it's establishment). Within two years of Alexander's death the Wars of the Diadochi occurred between the fractured empire and it ended with many different kingdoms being established.

EDIT: The empire was originally partitioned between his generals at the Partition of Babylon in 323BC and after 321BC there was little unity between the different generals kingdoms