The number of charged particles inside jets is a widely used discriminant for identifying the quark or gluon nature of the initiating parton and is sensitive to both the perturbative and non-perturbative components of fragmentation. This paper presents a measurement of the average number of charged particles with $p_\text{T}>500$ MeV inside high-momentum jets in dijet events using 20.3 fb$^{-1}$ of data recorded with the ATLAS detector in $pp$ collisions at $\sqrt{s}=8$ TeV collisions at the LHC. The jets considered have transverse momenta from 50 GeV up to and beyond 1.5 TeV. The reconstructed charged-particle track multiplicity distribution is unfolded to remove distortions from detector effects and the resulting charged-particle multiplicity is compared to several models. Furthermore, quark and gluon jet fractions are used to extract the average charged-particle multiplicity for quark and gluon jets separately.