We present measurements of radioactive contamination in the high-resistivity silicon charge-coupled devices (CCDs) used by the DAMIC experiment to search for dark matter particles. Novel analysis methods, which exploit the unique spatial resolution of CCDs, were developed to identify \(\alpha\) and \(\beta\) particles. Uranium and thorium contamination in the CCD bulk was measured through \(\alpha\) spectroscopy, with an upper limit on the \(^{238}\)U (\(^{232}\)Th) decay rate of 5 (15) kg\(^{-1}\) d\(^{-1}\) at 95% CL. We also searched for pairs of spatially correlated electron tracks separated in time by up to tens of days, as expected from \(^{32}\)Si-\(^{32}\)P or \(^{210}\)Pb-\(^{210}\)Bi sequences of \(\beta\) decays. The decay rate of \(^{32}\)Si was found to be \(80^{+110}_{-65}\) kg\(^{-1}\) d\(^{-1}\) (95% CI). An upper limit of \(\sim\)35 kg\(^{-1}\) d\(^{-1}\) (95% CL) on the \(^{210}\)Pb decay rate was obtained independently by \(\alpha\) spectroscopy and the \(\beta\) decay sequence search. These levels of radioactive contamination are sufficiently low for the successful operation of CCDs in the forthcoming 100 g DAMIC detector.