Carbon nanotubes (CNTs) have emerged as promising materials for visible, infrared, and terahertz photodetectors. Further development of these photodetectors requires a fundamental understanding of the mechanisms that govern their behavior as well as the establishment of figures of merit for technology applications. Recently, a number of CNT detectors have been shown to operate based on the photothermoelectric effect. Here we present a figure of merit for these detectors, which includes the properties of the material and the device. In addition, we use a suite of experimental characterization methods for the thorough analysis of the electrical, thermoelectric, electrothermal, and photothermal properties of the CNT thin-film devices. Our measurements determine the quantities that enter the figure of merit, and allow us to establish a path towards future performance improvements.