We present an overview of the SkyMapper optical follow-up programme for gravitational-wave event triggers from the LIGO/Virgo observatories, which aims at identifying early GW170817-like kilonovae out to $\sim200\,\mathrm{Mpc}$ distance. We describe our robotic facility for rapid transient follow-up, which can target most of the sky at $\delta<+10\deg $ to a depth of $i_\mathrm{AB}\approx 20\,\mathrm{mag}$ . We have implemented a new software pipeline to receive LIGO/Virgo alerts, schedule observations and examine the incoming real-time data stream for transient candidates. We adopt a real-bogus classifier using ensemble-based machine learning techniques, attaining high completeness ( $\sim98\%$ ) and purity ( $\sim91\%$ ) over our whole magnitude range. Applying further filtering to remove common image artefacts and known sources of transients, such as asteroids and variable stars, reduces the number of candidates by a factor of more than 10. We demonstrate the system performance with data obtained for GW190425, a binary neutron star merger detected during the LIGO/Virgo O3 observing campaign. In time for the LIGO/Virgo O4 run, we will have deeper reference images allowing transient detection to $i_\mathrm{AB}\approx 21\,\mathrm{mag}$ .