The recent development of petawatt-class lasers with kilojoule-picosecond pulses, such as OMEGA EP [L. Waxer et al., Opt. Photonics News 16, 30 (2005)], provides a new diagnostic capability to study inertial-confinement-fusion (ICF) and high-energy-density (HED) plasmas. Specifically, petawatt OMEGA EP pulses have been used to backlight OMEGA implosions with energetic proton beams generated through the target normal sheath acceleration (TNSA) mechanism. This allows time-resolved studies of the mass distribution and electromagnetic field structures in ICF and HED plasmas. This principle has been previously demonstrated using Vulcan to backlight six-beam implosions [A. J. Mackinnon et al., Phys. Rev. Lett. 97, 045001 (2006)]. The TNSA proton backlighter offers better spatial and temporal resolution but poorer spatial uniformity and energy resolution than previous D(3)He fusion-based techniques [C. Li et al., Rev. Sci. Instrum. 77, 10E725 (2006)]. A target and the experimental design technique to mitigate potential problems in using TNSA backlighting to study full-energy implosions is discussed. The first proton radiographs of 60-beam spherical OMEGA implosions using the techniques discussed in this paper are presented. Sample radiographs and suggestions for troubleshooting failed radiography shots using TNSA backlighting are given, and future applications of this technique at OMEGA and the NIF are discussed.