Charged particles from the hadronic final state of electron-proton deep inelastic scattering at HERA are measured by the Central Tracking Detector (CTD), a component of the ZEUS detector. The operation of the CTD is described. Using the incomplete tracking data from 1992 and the first data, taken during 1993, for which the CTD was fully instrumented, some systematic effects on the measurement of charged particles are investigated, and where possible, corrected. Factors affecting assignment of tracks to the main event vertex are discussed. The angular and momentum ranges which are reliable for physics analysis are identified. Momentum flow of charged hadrons, in pseudorapidity and in azimuth, is measured in the HERA laboratory frame with respect to the nominal direction of the struck quark, which is obtained from the reconstructed event kinematics. Comparisons are made with the predictions of three Monte Carlo models. Monte Carlo techniques are used to correct the data for detector effects. The reconstructed momentum four-vectors of the charged hadrons are transformed to the Breit frame, where the momentum flow versus pseudorapidity is measured. Scaled momentum spectra of final state hadrons associated, in the Breit frame, with the fragmentation region of the struck quark current region), are measured. The evolution of multiplicity and scaled momentum distributions with increasing virtuality Q of the mediating photon are shown and compared with data from e+e- annihilation experiments. The increased effect at low x-Q2 of initial-state QCD radiation in DIS is evident both in the momentum flow distributions and in the Breit-frame evolution of current region multiplicity with Q, but appears not to significantly affect the scaled momentum evolution which is found to be consistent with e+e- results.