In the Marvel Cinematic Universe, Natasha Romanov (AKA Natalia Romanov, AKA "Black Widow") is not the product of the enhanced serum but rather a Soviet programme designed to take promising young girls and train them to become anti-Western assassins.
We see the Soviet spy training programme (confirmed in an interview to be the precursor to the Black Widow programme) in Agent Carter. This regime includes brainwashing, physical and mental abuse, fight training, marksmanship and the killing of live targets (and each other) in order to demonstrate worthiness to 'graduate'.
Under the cover of a ballet school, Natalia was trained as an assassin, then sent on a series of missions culminating in her defection to the west, whereupon she became a member of S.H.I.E.L.D.
Her talents and abilities can be best described as peak human rather than actually superhuman.