We use the high-resolution cosmological simulation Illustris to investigate the clustering of supermassive black holes across cosmic time, the link between black hole clustering and host halo masses, and the implications for black hole duty cycles. Our predicted black hole correlation length and bias match the observational data very well across the full redshift range probed. Black hole clustering is strongly luminosity dependent on small, 1-halo scales, with some moderate dependence on larger scales of a few Mpc at intermediate redshifts. We find black hole clustering to evolve only weakly with redshift, initially following the behaviour of their hosts. However, below z ~ 2 black hole clustering increases faster than that of their hosts, which leads to a significant overestimate of the clustering-predicted host halo mass. The full distribution of host halo masses is very wide, including a low-mass tail extending up to an order of magnitude below the naive prediction for minimum host mass. Our black hole duty cycles, $f$duty, follow a power-law dependence on black hole mass and decrease with redshift, and we provide accurate analytic fits to these. The increase in clustering amplitude at late times, however, means that duty cycle estimates based on black hole clustering can overestimate $f$duty substantially, by more than two orders of magnitude. We find the best agreement when the minimum host mass is assumed to be 10$\mathrm{<mrow\; data-mjx-texclass="ORD">11.2</mrow>}$M⊙, which provides an accurate measure across all redshifts and luminosity ranges probed by our simulation.