Using a sample of (10087 +/- 44) x 10(6) J/psi events accumulated with the BESIII detector, we analyze the decays eta -> pi(+) pi(-) l(+) l(-) (l = e or mu) via the process J/psi -> gamma eta. The branching fraction of eta -> pi(+) pi(-) e(+) e(-) is measured to be B(eta -> pi(+) pi(-) e(+) e(-)) = (3.07 +/- 0.12(stat) +/- 0.19(syst)) x 10(-4). No signal events are observed for the eta -> pi(+) pi(-) mu(+) mu(-) decay, leading to an upper limit on the branching fraction of B(eta -> pi(+) pi(-) mu(+) mu(-)) < 4.0 x 10(-7) at the 90% confidence level. Furthermore, the CP-violation asymmetry parameter is found to be A(CP)(eta -> pi(+) pi(-) e(+) e(-)) = (-4.04 +/- 4.69(stat) +/- 0.14(syst))%, showing no evidence of CP-violation with current statistics. Additionally, we extract the transition form factor from the decay amplitude of eta -> pi(+) pi(-) e(+) e(-). Finally, axionlike particles are searched for via the decay eta -> pi(+) pi(-) a, a -> e(+) e(-), and upper limits on this branching fraction relative to that of eta -> pi(+) pi(-) e(+) e(-) are presented as a function of the axionlike particle mass in the range 5-200 MeV/c(2).