This article presents the measurement of charged-particle pseudorapidity ($\eta$) density, $\mathrm{d}N_{\rm ch}/\mathrm{d}\eta$, in proton$-$proton (pp) collisions at a centre-of-mass energy $\sqrt{s}$ = 13.6 TeV, and in lead$-$lead (Pb$-$Pb) collisions at a centre-of-mass energy per nucleon pair $\sqrt{s_{\mathrm{NN}}}$ = 5.36 TeV. The analysis is performed using the Run 3 data recorded during 2022 and 2023 by the upgraded ALICE detector. The charged-particle multiplicity is measured at midrapidity ($|\eta|<1$) using the new monolithic active pixel sensors-based Inner Tracking System and the Time Projection Chamber upgraded with Gas Electron Multiplier-based readout system. The measurements in pp collisions are reported for inelastic events with at least one charged particle having $|\eta|<1$ whereas for Pb$-$Pb collisions, the $\mathrm{d}N_{\rm ch}/\mathrm{d}\eta$ is obtained for different centrality classes, ranging from 0$-$5\% (most central) to 70$-$80\% (most peripheral). The energy dependence of average charged-particle pseudorapidity density ($\langle \mathrm{d}N_{\rm ch}/\mathrm{d}\eta \rangle$) measured in $|\eta|<0.5$ is studied and compared to earlier measurements at lower collision energies. In Pb$-$Pb collisions, the evolution of $\langle \mathrm{d}N_{\rm ch}/\mathrm{d}\eta \rangle$ as a function of the average number of participating nucleons, $\langle N_{\mathrm{part}} \rangle$, determined with a Glauber model, is also studied and compared with predictions from theoretical models.