Magnetic properties of the substitution series Ru$_{1-x}$Cr$_x$Cl$_3$ were investigated to determine the evolution from the anisotropic Kitaev magnet $alpha$-RuCl$_3$ with $J_{rm eff} = 1/2$ magnetic Ru$^{3+}$ ions to the isotropic Heisenberg magnet CrCl$_3$ with $S = 3/2$ magnetic Cr$^{3+}$ ions. Magnetization measurements on single crystals revealed a reversal of the magnetic anisotropy under doping, which we argue to arise from the competition between anisotropic Kitaev and off-diagonal interactions on the Ru-Ru links and approximately isotropic Cr-Ru and isotropic Cr-Cr interactions. In addition, combined magnetization, ac susceptibility and specific-heat measurements clearly show the destabilization of the long-range magnetic order of $alpha$-RuCl$_3$ in favor of a spin-glass state of Ru$_{1-x}$Cr$_x$Cl$_3$ for a low doping of $xbacksimeq0.1$. The corresponding freezing temperature as a function of Cr content shows a broad maximum around $xbacksimeq0.45$.
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