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), using data collected with the Collider Detector at Fermilab. With an integrated luminosity of 9.1 fb (sub -1), the leptonic forward-backward asymmetry, A(sub l)(sup FB), is measured to be 0.072 plus or minus 0.060 and the leptonic pair forward-backward asymmetry, A(sub ll)(sup FB), is measured to be 0.076 plus or minus 0.082, compared with the standard model predictions of A(sub l)(sup FB)=0.038 plus or minus 0.003 and A(sub ll)(sup FB) FB = 0.038 0.003 and A(sub ll)(sup FB)=0.48 plus or minus 0.004, respectively. Additionally, we combine the A(sub l(sup FB)) result with a previous determination from a final state with a single lepton and hadronic jets and obtain A(sub l(sup FB)=0.020(sub plus 0.028 minus 0.026)).