The Cherenkov Telescope Array (CTA) is the next-generation gamma-ray observatory that is expected to reach one order of magnitude better sensitivity than that of current telescope arrays. The Large-Sized Telescopes (LSTs) have an essential role in extending the energy range down to 20 GeV. The prototype LST (LST-1) proposed for CTA was built in La Palma, the northern site of CTA, in 2018. LST-1 is currently in its commissioning phase and moving towards scientific observations. The LST-1 camera consists of 1855 photomultiplier tubes (PMTs) which are sensitive to Cherenkov light. PMT signals are recorded as waveforms sampled at 1 GHz rate with Domino Ring Sampler version 4 (DRS4) chips. Fast sampling is essential to achieve a low energy threshold by minimizing the integration of background light from the night sky. Absolute charge calibration can be performed by the so-called F-factor method, which allows calibration constants to be monitored even during observations. A calibration pipeline of the camera readout has been developed as part of the LST analysis chain. The pipeline performs DRS4 pedestal and timing corrections, as well as the extraction and calibration of charge and time of pulses for subsequent higher-level analysis. The performance of each calibration step is examined, and especially charge and time resolution of the camera readout are evaluated and compared to CTA requirements. We report on the current status of the calibration pipeline, including the performance of each step through to signal reconstruction, and the consistency with Monte Carlo simulations.