The dynamics of the microlayer beneath a growing bubble in nucleate boiling significantly impacts the heat-transfer characteristics of the process. The minute thickness of the microlayer motivates the use of direct numerical simulation (DNS) to model its behaviour if empirical models are to be avoided. In this work, we develop a computational strategy for utilising DNS to model nucleate boiling by resolving explicitly the microlayer, directly coupling, in a stable manner, the mass, momentum and energy conservation equations with the conjugate heat transfer between the solid and fluid domains. To this end, closure models for the treatment of interfacial heat transfer and the dynamic contact angle are introduced and substantiated. The computational procedure is validated against relevant experimental data recently measured at the Massachusetts Institute of Technology; it is shown that the main observed growth features and surface heat-transfer characteristics are well reproduced using our model. We go on to perform a sensitivity study of the dependence of the initial microlayer thickness distribution on the applied superheat and fluid properties. The results indicate that an equation derived from lubrication theory captures the observed trends well. Finally, a first demonstration of DNS of boiling with an explicitly resolved microlayer in three-dimensional Cartesian coordinates is presented in one of the appendices.