There is only one taxonomic "reality" and that is the species (and even that can be discussed!). All other Linnean "ranks" are completely arbitrary, depending on the author's opinion. A phylogenetic (cladistic) analysis produces many monophyletic species groups (called taxa) on which the Linnean classification cannot be applied. Conclusion: forget Linnaeus and keep to the neutral term "taxon". It works very well! If you absolutely want to use Linnean ranks, you should stick to the Code of Zoological Nomenclature (http://www.iczn.org/iczn/index.jsp).
To restate your question, I presume that you are asking about the process of classifying an organism from scratch as opposed to asking about what the different taxonomic ranks are.
As far as the process, there are two general systems, one based on common attributes and one based genetically. While a genetic basis would be more accurate to required for lower taxa (family, genus, and species), broader physical traits can be used for most of the higher taxa (domain, kingdom, phylum, class, order).
Taxonomic ranks represent monophyletic groupings of species, so the traits used must represent genetic relatedness. However, a taxa within one group (i.e. Class Insecta) may not always represent the same level of relatedness as within another group (i.e. Class Mollusca).
One problem in classification and taxonomy is that many disagreements exist on what constitutes a particular taxonomic rank. Some are more obvious (i.e. domain being based on nuclear structure and ribosomal RNA or species being based on the ability to reproduce viable offspring) while others are rather questionable (i.e. order or family).
There is only one taxonomic "reality" and that is the species (and even that can be discussed!). All other Linnean "ranks" are completely arbitrary, depending on the author's opinion. A phylogenetic (cladistic) analysis produces many monophyletic species groups (called taxa) on which the Linnean classification cannot be applied. Conclusion: forget Linnaeus and keep to the neutral term "taxon". It works very well! If you absolutely want to use Linnean ranks, you should stick to the Code of Zoological Nomenclature (http://www.iczn.org/iczn/index.jsp).
Oh, by the way: I am afraid Simpson will not be of great help. He was a strong opponent (together with E. Mayer) of Hennig and of the cladistic system. Hence he does not incorporate the cladistic taxa into the nomenclature as we should do today.