We investigate the cosmological implications of studying galaxy clustering using a tomographic approach applied to the final Baryon Oscillation Spectroscopic Survey (BOSS) DR12 galaxy sample, including both auto- and cross-correlation functions between redshift shells. We model the signal of the full shape of the angular correlation function, ω(θ), in redshift bins using state-of-the-art modelling of non-linearities, bias and redshift-space distortions. We present results on the redshift evolution of the linear bias of BOSS galaxies, which cannot be obtained with traditional methods for galaxy-clustering analysis. We also obtain constraints on cosmological parameters, combining this tomographic analysis with measurements of the cosmic microwave background (CMB) and Type Ia supernova (SNIa). We explore a number of cosmological models, including the standard Λ cold dark matter model and its most interesting extensions, such as deviations from wDE = −1, non-minimal neutrino masses, spatial curvature and deviations from general relativity (GR) using the growth-index γ parametrization. These results are, in general, comparable to the most precise present-day constraints on cosmological parameters, and show very good agreement with the standard model. In particular, combining CMB, ω(θ) and SNIa, we find a value of wDE consistent with −1 to a precision better than 5 per cent when it is assumed to be constant in time, and better than 6 per cent when we also allow for a spatially curved Universe.