Laminar counterflow diffusion flames provide enhanced understanding of fundamentals for non-premixed turbulent combustion. Current studies have mainly focused on lower or moderate pressure conditions, yet studies at supercritical pressures are less well-documented. This paper systematically investigates a counterflow diffusion flame of the oxygen/methane mixture at both subcritical and supercritical pressures in a wide range of flow strain rates and injection temperatures. We employ fundamental thermodynamic theories and a real-fluid equation of state to evaluate thermophysical properties and utilize the flame-controlling continuation method to capture the extinction strain rate. The influences of the flow strain rate, pressure, and inlet temperature of oxygen on flame structures, heat release rates, and extinction characteristics are discussed in detail. The results indicate that the flame thickness and the global heat release rate are closely related to the pressure-weighted strain rate and that the extinction strain rate linearly increases with pressure as the pressure is lower than 50 atm, but starts to bend over nonlinearly when the pressure is higher than 50 atm. The injection temperature of oxygen negligibly affects flame characteristics but plays a crucial role in accurately computing thermophysical properties in the transcritical zone. The results can be used for the establishment of tabulated chemistry library in supercritical combustion modeling.