  Kerr black holes with scalar hair are solutions of the Einstein-Klein-Gordon
field equations describing regular (on and outside an event horizon),
asymptotically flat black holes with scalar hair (arXiv:1403.2757). These black
holes interpolate continuously between the Kerr solution and rotating boson
stars in D=4 spacetime dimensions. Here we provide details on their
construction, discussing properties of the ansatz, the field equations, the
boundary conditions and the numerical strategy. Then, we present an overview of
the parameter space of the solutions, and describe in detail the space-time
structure of the black holes exterior geometry and of the scalar field for a
sample of reference solutions. Phenomenological properties of potential
astrophysical interest are also discussed, and the stability properties and
possible generalizations are commented on. As supplementary material to this
paper we make available numerical data files for the sample of reference
solutions discussed, for public use.
