Energy levels, magnetic dipole, and electric quadrupole hyperfine structure of the superheavy element rutherfordium (Rf, Z=104) and its first three ions are calculated using a combination of the configuration interaction, coupled-cluster single-doubles, and many-body perturbation theory techniques. The results are to be used in future interpretations of the measurements in terms of nuclear magnetic dipole and electric quadrupole momenta. To have a guide on the accuracy of the study, we perform similar calculations for hafnium (Hf, Z=72) and its ions. Hf is a lighter analog of Rf with a similar electronic structure. Good agreement with the experiment for Hf and with available previous calculations of the energy levels of Rf is demonstrated.