The width and shape of the density of states (DOS) are key parameters to describe the charge transport of organic semiconductors. Here we extract the DOS using scanning Kelvin probe microscopy on a self-assembled monolayer field effect transistor (SAMFET). The semiconductor is only a single monolayer which has allowed extraction of the DOS over a wide energy range, pushing the methodology to its fundamental limit. The measured DOS consists of an exponential distribution of deep states with additional localized states on top. The charge transport has been calculated in a generic variable range-hopping model that allows any DOS as input. We show that with the experimentally extracted DOS an excellent agreement between measured and calculated transfer curves is obtained. This shows that detailed knowledge of the density of states is a prerequisite to consistently describe the transfer characteristics of organic field effect transistors.