The decays chi(c1) -> J/psi mu(+)mu(-) and chi(c1) -> J/psi mu(+)mu(-) are observed and used to study the resonance parameters of the chi(c1) and chi(c2) mesons. The masses of these states are measured to be m(chi(c1)) = 3510.71 +/- 0.04(stat) +/- 0.09(syst) MeV and m(chi(c2)) = 3556.10 +/- 0.06(stat) +/- 0.11(syst) MeV, where the knowledge of the momentum scale for charged particles dominates the systematic uncertainty. The momentum-scale uncertainties largely cancel in the mass difference m(chi(c2)) - m(chi(c1)) = 45.39 +/- 0.07(stat) +/- 0.03(syst) MeV. The natural width of the chi(c2) meson is measured to be Gamma(chi(c2)) = 2.10 +/- 0.20(stat) +/- 0.02(syst) MeV. These results are in good agreement with and have comparable precision to the current world averages.