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.
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