In earlier days, biological classification systems were often described as artifical or natural, with natural systems reflecting the 'real' relationships among living beings, and artificial ones allowing classification only for some limited purpose (Gilmour, 1937). Natural systems were supposed to be based on a large number of characters, with a focus on characters 'fundamental' to organisms. Artifical systems were supposed to be based on fewer, arbitrarily chosen characters.
As an example, consider Linnaeus' works on the classification of plants. Linnaeus' Systema Naturae (1735) classified plants based on floral morphology alone. However, his later work Genera Plantarum (1737) included more characters, such as the morphology of fruits. Linnaeus therefore regarded the latter classification as being more natural. (See the Encyclopedia Britannica entry on Linnaeus for a more detailed discussion of this.)
As for the two-kingdom classification, I couldn't find any sources calling it either artificial or natural.
However, the natural vs artificial dichotomy is not always straightforward. For instance, how do we know which characters are 'fundamental' and will help us develop a natural classification system? To quote Judd et al (2008):
For hundreds of years, botanists tried to develop classifications that were "natural"... Unfortunately, the word natural has no fixed meaning; rather, authors have used it to mean something that agrees with their own ideas about nature or about constructing classifications or systems. Eighteenth-century systematists had ideas about nature that were very different from ours—they were certainly not evolutionary—and their systematic practice and classifications are best interpreted in terms of how they understood nature.
Thus, in modern biology, 'natural classification' and 'artificial classification' are better regarded as historical ideas than as useful descriptors. The focus is now on classification that integrates an understanding of evolutionary relationships, i.e. phylogenetic classification.