The need for improved engine efficiencies has motivated the development of high-pressure combustion systems, in which operating conditions achieve and exceed critical conditions. Associated with these conditions are strong variations in thermo-transport properties as the fluid undergoes mixing and phase transition, and two-stage ignition with low-temperature combustion. Accurately simulating these physical phenomena at real-fluid environments remains a challenge. This study examines a diffuse-interface method for simulating the injection and ignition of n-dodecane at transcritical conditions. To this end, a compressible solver with a real-fluid state equation and finite-rate chemistry is employed. Simulations of an ECN-relevant diesel-fuel injector are performed for both inert and reacting conditions. For the spray ignition, four specific operating points (corresponding to ambient temperatures between 900 K and 1200 K) are investigated to examine effects of the real-fluid environment and low-temperature chemistry. Comparisons with available experimental data demonstrate that the presented numerical method adequately captures the diesel fuel injection and auto-ignition processes under transcritical conditions.