As the electron beam travels through the klystron, the RF cavities in klystron bunches and provides effective velocity modulation to the beam. Interaction between the electron beam and the RF fields occurring within the RF section of the klystron is responsible for the overall gain of the amplifier. Design of the RF cavity involves optimization of several figures of merit including resonant frequency, shunt impedance, quality factor, power dissipation, R/Q ratio amongst others. The re-entrant feature in the cavity aids in concentrating electric field in the RF gap region and hence enhancing the velocity modulation of electron beam. Various designs of the RF cavity for an S-band klystron have been modelled and analysed to optimise the figures of merit, while minimising the peak electric field within Kilpatrick limit. Study of tuner efficacy in individual cavities has been carried out for determining the overall tuning range. Error sensitivity analysis has been done to study effect of individual geometrical parameter on the resonant frequency. Temperature rise and heat load on the RF cavities were calculated. This serves as an input for the cooling mechanism design. In this paper, a detailed multi-physics coupled study and design optimisation of the RF section of klystron has been presented.