Aims: We present constraints on the thermodynamical structure of the chromosphere from ground-based observations of the Ca ii H line profile near and off the solar limb. Methods: We obtained a slit-spectrograph data set of the Ca ii H line with a high signal-to-noise ratio in a field of view extending 20'' across the limb. We analyzed the spectra for the characteristic properties of average and individual off-limb spectra. We used various tracers of the Doppler shifts, such as the location of the absorption core, the ratio of the two emission peaks H2V and H2R, and intensity images at a fixed wavelength. Results: The average off-limb profiles show a smooth variation with increasing limb distance. The line width increases up to a height of about 2 Mm above the limb. The profile shape is fairly symmetric with nearly identical H2V and H2R intensities; at a height of 5 Mm, it changes into a single Gaussian without emission peaks. We find that all off-limb spectra show large Doppler shifts that fluctuate on the smallest resolved spatial scales. The variation is more prominent in cuts parallel to the solar limb than on those perpendicular to it. As far as individual structures can be unequivocally identified at our spatial resolution, we find a specific relation between intensity enhancements and Doppler shifts: elongated brightenings are often flanked all along their extension by velocities in opposite directions. Conclusions: The average off-limb spectra of Ca ii H present a good opportunity to test static chromospheric atmosphere models because they lack the photospheric contribution that is present in disk-center spectra. We suggest that the observed relation between intensity enhancements and Doppler shifts could be caused by waves propagating along the surfaces of flux tubes: an intrinsic twist of the flux tubes or a wave propagation inclined to the tube axis would cause a helical shape of the Doppler excursions, visible as opposite velocity at the sides of the flux tube. Spectroscopic data allow one to distinguish this from a sausage-mode oscillation where the maximum Doppler shift and the tube axis would coincide. Appendices are available in electronic form at http://www.andaa.orgThe Data set is available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/531/A173