Check the CASSCF option in Gaussian, especially the Examples tab which shows output from a spin-orbit calculation: http://gaussian.com/cas/?tabid=2#CASSCF_keyword__SpinOrbit_option

Simply calculate an open-shell system (single point task is OK. No any special keyword is needed), at the end of output file you can find "Isotropic Fermi Contact Couplings" section, the values under "a.u." are simply spin density at position of each nucleus, which can be very easily converted to Fermi contact term of hyperfine coupling, see J. Phys. Chem. A, 101, 3174 (1997) for more information.

By the way, in order to get reliable result, you need to use a basis set with tight functions, such as IGLO-III and pc-J series of basis sets, which are not build-in basis sets in Gaussian but can be obtained from BSE basis set library.

Thank you so much