We measure the inclusive forward-backward asymmetry of the charged-lepton pseudorapidities from top-quark pairs produced in proton-antiproton collisions and decaying to final states that contain two charged leptons (electrons or muons). The data are collected with the Collider Detector at Fermilab and correspond to an integrated luminosity of 9.1 fb(-1). We measure the leptonic forward-backward asymmetry, A(FB)(ℓ), to be 0.072 ± 0.060 and the leptonic pair forward-backward asymmetry, A(FB)(ℓℓ), to be 0.076 ± 0.082. The measured values can be compared with the standard model predictions of A(FB)(ℓ) = 0.038 ± 0.003 and A(FB)(ℓℓ) = 0.048 ± 0.004, respectively. Additionally, we combine the A(FB)(ℓ) result with a previous determination from a final state with a single lepton and hadronic jets and obtain A(FB)(ℓ) = 0.090(-0.026)(+0.028).