Pulsars are highly magnetised, rapidly rotating neutron stars most frequently observed in radio wavelengths as their emitted beam path crosses an observer's line-of-sight. Millisecond pulsars (MSPs) demonstrate exceptional rotational stability over long timescales, rivalling the accuracy of the best terrestrial atomic clocks. Pulsar timing investigations can uncover a wealth of knowledge when high-precision, such as 1 microsecond or better, is achieved. In this thesis we present the novel approach of using a combination of state-of-the-art, high-precision pulsar timing tools for polarimetric calibration, mitigation of radio frequency interference (RFI), pulse time-of-arrival (TOA) analysis, and post processing into one pulsar processing pipeline (PSRPL). We have integrated the CoastGuard algorithm for RFI excision, performed instrumental calibration via Measurement Equation Template Matching (METM) followed by the Matrix Template Matching (MTM) algorithm for producing TOA estimates, and analysed the resulting TOAs with Tempo2 and Temponest. Our method has been applied to a sample of five pulsars that are highly susceptible to calibration errors, as predicted by van Straten (2013): PSR J0437-4715, PSR J1022+1001, PSR J1045-4509, PSR J1600-3053, and PSR J1643-1224. Approximately 8 years of historical, observational data were analysed from the Parkes 64-m radio telescope's CASPSR backend (or instrument) for each pulsar in our sample. We have improved the timing residuals of all MSPs in our sample (e.g. achieving 60-nanosecond timing residuals for PSR J0437-4715), with four out of five better than predicted, and shown that PSRPL is the optimal pipeline for high-precision pulsar timing over those using conventional methods (e.g. the Ideal Feed Assumption (IFA) and Scalar Template Matching (STM) algorithms). This result is an important step in the search for low frequency (nHz) Gravitational Waves (GWs) using Pulsar Timing Arrays (PTAs). We conclude by discussing possible further improvements to PSRPL and its intended application for both new and historical data from Parkes PTA instruments, as well as international telescopes such as MeerKAT, as we have demonstrated that PSRPL can improve timing results.