The precision of the Gaia data offers a unique opportunity to study the internal velocity field of star clusters. We develop and validate a forward-modelling method for the internal motions of stars in a cluster. The model allows an anisotropic velocity dispersion matrix and linear velocity gradient describing rotation and shear, combines radial velocities available for a subset of stars, and accounts for contamination from background sources via a mixture model. We apply the method to Gaia DR2 data of the Hyades cluster and its tidal tails, dividing and comparing the kinematics of stars within and beyond 10pc, which is roughly the tidal radius of the cluster. While the velocity dispersion for the cluster is nearly isotropic, the velocity ellipsoid for the tails is clearly elongated with the major axis pointing towards the Galactic centre. We find positive and negative expansions at ~2{sigma} significance in Galactic azimuthal and vertical directions for the cluster but no rotation. The tidal tails are stretching in a direction tilted from the Galactic centre while equally contracting as the cluster in Galactic vertical direction. The tails have a shear (A) of 16.90+/-0.92m/s/pc and a vorticity (B) of -6.48+/-1.15m/s/pc, values distinct from the local Oort constants. By solving the Jeans equations for flattened models of the Hyades, we show that the observed velocity dispersions are a factor of ~2 greater than required for virial equilibrium due to tidal heating and disruption. From simple models of the mass loss, we estimate that the Hyades is close to final dissolution with only a further ~<30Myr left.