Context. The star cluster R136 inside the Large Magellanic Cloud hosts a rich population of massive stars, including the most massive stars known. The strong stellar winds of these very luminous stars impact their evolution and the surrounding environment. We currently lack detailed knowledge of the wind structure that is needed to quantify this impact. Aims. Our goal is to observationally constrain the stellar and wind properties of the massive stars in R136, in particular the wind-structure parameters related to wind clumping. Methods. We simultaneously analyse optical and ultraviolet spectroscopy of 53 O-type and three WNh-stars using the Fastwind model atmosphere code and a genetic algorithm. The models account for optically thick clumps and effects related to porosity and velocity-porosity, as well as a non-void interclump medium. Results. We obtain stellar parameters, surface abundances, mass-loss rates, terminal velocities, and clumping characteristics and compare them to theoretical predictions and evolutionary models. The clumping properties include the density of the interclump medium and the velocity-porosity of the wind. For the first time, these characteristics are systematically measured for a wide range of effective temperatures and luminosities. Conclusions. We confirm a cluster age of 1.0–2.5 Myr and derived an initial stellar mass of ≥250 M_⊙ for the most massive star in our sample, R136a1. The winds of our sample stars are highly clumped, with an average clumping factor of f_cl = 29 ± 15. We find tentative trends in the wind-structure parameters as a function of the mass-loss rate, suggesting that the winds of stars with higher mass-loss rates are less clumped. We compare several theoretical predictions to the observed mass-loss rates and terminal velocities and find that none satisfactorily reproduce both quantities. The prescription of Krtička & Kubát (2018) matches the observed mass-loss rates best.