We perform a joint determination of the distance-redshift relation and cosmic expansion rate at redshifts z = 0.44, 0.6 and 0.73 by combining measurements of the baryon acoustic peak and Alcock-Paczynski distortion from galaxy clustering in the WiggleZ Dark Energy Survey, using a large ensemble of mock catalogues to calculate the covariance between the measurements. Further combining our results with other baryon acoustic oscillation and distant supernovae datasets, we use a Monte Carlo Markov Chain technique to determine the evolution of the Hubble parameter H(z) as a stepwise function in 9 redshift bins of width dz = 0.1, also marginalizing over the spatial curvature. Our measurements of H(z), which have precision better than 7% in most redshift bins, are consistent with the expansion history predicted by a cosmological-constant dark-energy model, in which the expansion accelerates at redshift z < 0.7. We also measure the normalized cosmic growth rate at z = 0.44, 0.6 and 0.73, together with its covariance with the expansion history, using redshift-space distortions in the WiggleZ Survey dataset. The measured growth rate is consistent with the same cosmological-constant model that describes the expansion history.