  Context: The star cluster R136 inside the LMC 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: To observationally constrain the stellar
and wind properties of the massive stars in R136, in particular the parameters
related to wind clumping. Methods: We simultaneously analyse optical and UV
spectroscopy of 53 O-type and 3 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
these 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 derive
an initial stellar mass of $\geq 250 {\rm M}_\odot$ for the most massive star
in our sample, R136a1. The winds of our sample stars are highly clumped, with
an average clumping factor of $f_{\rm cl}=29\pm15$. We find tentative trends in
the wind-structure parameters as a function of 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 reproduces both quantities. The
prescription of Krti\v{c}ka & Kub\'at (2018) matches best the observed
mass-loss rates.
