Recent discovery of topological superconductors (TSCs) has sparked enormous interest. Realization of TSC requires a delicate tuning of multiple microscopic parameters, which remains a great challenge. Here, we develop a first-principles approach to quantify realistic conditions of TSC by solving self-consistently Bogoliubov-de Gennes equation based on Wannier function construction of band structure, in presence of Rashba spin-orbit coupling, Zeeman splitting and electron-phonon coupling. We further demonstrate the power of this new method by predicting the Mn-doped GeTe (Ge$_{1-x}$Mn$_x$Te) monolayer - a well-known dilute magnetic semiconductor showing superconductivity under hole doping - to be a Class D TSC with Chern number of -1 and chiral Majorana edge modes. By constructing a first-principles phase diagram in the parameter space of temperature and Mn concentration, we propose the TSC phase can be induced at a lower-limit transition temperature of ~40 mK and the Mn concentration of $x$~0.015%. Our approach can be generally applied to TSCs with a phonon-mediated pairing, providing useful guidance for future experiments.
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