The radio-frequency (RF) self-biased ion thruster is a neutraliser-free electric propulsion device in which a positive DC self-bias is established on the screen grid and electrons are periodically extracted during the negative half-cycles of the RF voltage, enabling the generation of a quasi-neutral plasma plume without an external neutraliser. However, with the introduction of an RF-biased screen grid and the self-bias effect, the plasma inside the discharge chamber exhibits more complex spatiotemporal behaviour than that in conventional ion thrusters. In particular, the sheath oscillation upstream of the ion optics and its influence on the screen-grid self-bias voltage remain insufficiently understood. This limits the applicability of existing self-bias theories based on a time-independent sheath potential drop and hinders the physics-based optimisation of RF self-biased ion thrusters. In this work, the plasma dynamics inside the discharge chamber of an RF self-biased ion thruster are systematically investigated using a two-dimensional axisymmetric implicit particle-in-cell/Monte Carlo collision (PIC/MCC) model. The model includes a dielectric discharge chamber, an inductively coupled RF coil, an RF-biased screen grid, a grounded accelerator grid, and a grounded thruster shell. The spatiotemporal distributions of the electron and ion densities, plasma potential, chamber-wall potential, and currents collected by the screen grid and chamber wall are analysed over one screen-grid RF period. The physical origin of the sheath oscillation is further examined through electrical topology analysis. The simulation results show that the sheaths near the screen grid and the inner chamber wall oscillate alternately within the RF cycle. Electrons periodically migrate between the upstream region and the screen-grid side, whereas ions respond only weakly to the high-frequency screen-grid voltage. Therefore, the oscillation is mainly concentrated in the sheath regions, while the bulk plasma remains relatively weakly modulated. The plasma potential, screen-grid potential, and chamber-wall potential oscillate at the same frequency but with different amplitudes and DC offsets, indicating that the plasma potential does not vary synchronously with the screen-grid voltage. The screen-grid sheath exhibits a large potential drop containing a significant RF component, whereas the RF component of the chamber-wall sheath potential drop is much smaller. Meanwhile, the electron current collected by the screen grid shows a distinct pulsed behaviour, while the ion current varies more smoothly. These features are similar to those observed in asymmetric capacitively coupled plasma discharges, although the dominant plasma heating mechanism in the present thruster remains inductive coupling. The sheath oscillation mechanism can be understood from the electrical topology of the discharge chamber. The chamber inner wall can be approximately regarded as an equipotential surface connected to ground through the chamber-shell stray capacitance, while the screen grid acts as a small-area RF electrode. The applied RF voltage is therefore divided among the screen-grid sheath capacitance, the chamber-wall sheath capacitance, and the stray capacitance. Because the screen-grid sheath capacitance is smaller than the chamber-wall sheath capacitance and is comparable to the stray capacitance, a larger RF voltage drop develops across the screen-grid sheath, resulting in a stronger sheath oscillation near the screen grid. Finally, the influence of sheath oscillation on the screen-grid selfbias voltage is clarified. A stronger sheath oscillation increases the RF component of the screen-grid sheath potential drop and consequently lowers the steady-state DC self-bias voltage on the screen grid. Therefore, weakening the sheath oscillation is beneficial for increasing the screen-grid self-bias voltage and improving the ion acceleration capability. Based on this mechanism, reducing the chamber-shell stray capacitance, for example by enlarging the gap between the discharge chamber and the thruster shell, is proposed as a feasible optimisation strategy. This study reveals the sheath oscillation characteristics and formation mechanism inside the discharge chamber of RF self-biased ion thrusters, and provides a theoretical basis for their structural optimisation and performance improvement of RF self-biased ion thrusters.