Hexagonal $\alpha$-Ru trichloride single crystals exhibit a strong magnetic anisotropy and we show that upon applying fields up to 14 T in the honeycomb plane the successive magnetic order at T1=14K and T2=8K could be completely suppressed, whereas in the perpendicular direction the magnetic order is robust. Furthermore, the field dependence of $\chi$(T) implies coexisting ferro- and antiferromagnetic exchange between in-plane components of Ru3+ spins, whereas for out-of-plane components a strong antiferromagnetic exchange becomes evident. Ru101 zero-field nuclear magnetic resonance in the ordered state evidence a complex (probably noncoplanar chiral) long-range magnetic structure. The large orbital moment on Ru3+ is found in density-functional calculations.