Measurements of the top-antitop quark pair production charge asymmetry in the dilepton channel, characterized by two high-${p}_{\mathrm{T}}$ leptons (electrons or muons), are presented using data corresponding to an integrated luminosity of $20.3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ from $pp$ collisions at a center-of-mass energy $\sqrt{s}=8\text{ }\text{ }\mathrm{TeV}$ collected with the ATLAS detector at the Large Hadron Collider at CERN. Inclusive and differential measurements as a function of the invariant mass, transverse momentum, and longitudinal boost of the $t\overline{t}$ system are performed both in the full phase space and in a fiducial phase space closely matching the detector acceptance. Two observables are studied: ${A}_{\mathrm{C}}^{\ensuremath{\ell}\ensuremath{\ell}}$ based on the selected leptons and ${A}_{\mathrm{C}}^{t\overline{t}}$ based on the reconstructed $t\overline{t}$ final state. The inclusive asymmetries are measured in the full phase space to be ${A}_{\mathrm{C}}^{\ensuremath{\ell}\ensuremath{\ell}}=0.008\ifmmode\pm\else\textpm\fi{}0.006$ and ${A}_{\mathrm{C}}^{t\overline{t}}=0.021\ifmmode\pm\else\textpm\fi{}0.016$, which are in agreement with the Standard Model predictions of ${A}_{\mathrm{C}}^{\ensuremath{\ell}\ensuremath{\ell}}=0.0064\ifmmode\pm\else\textpm\fi{}0.0003$ and ${A}_{\mathrm{C}}^{t\overline{t}}=0.0111\ifmmode\pm\else\textpm\fi{}0.0004$.