The evolution of the polariton condensation threshold (P-thr) under incoherent optical pumping is investigated both theoretically and experimentally over a wide range of temperatures (4-340 K) and exciton-cavity photon detunings (-120-0 meV) in a multiple quantum-well GaN-based microcavity. The condensation phase diagram of these bosonic quasiparticles is first theoretically described within the framework of Bose-Einstein condensation of polaritons in the thermodynamic limit. Then a qualitative picture of cavity polariton relaxation kinetics including the impact of detuning and temperature is given before introducing a modeling of cavity polariton relaxation kinetics with semiclassical Boltzmann equations. The results of the theoretical modeling are finally compared with systematic measurements of P-thr. At low temperature and negative detunings, the polariton gas is far from thermal equilibrium and the condensation threshold is governed by the efficiency of the relaxation kinetics of the particles. Conversely, at high temperature and for less negative detunings, the relaxation kinetics is efficient enough to allow the achievement of a thermal polariton distribution function with a critical density for condensation given by the thermodynamic theory of Bose-Einstein condensation. For temperatures ranging between similar to 140 and 340 K, an optimum detuning is found experimentally, where the condensation threshold power is minimized. At high temperatures, polariton detrapping effects from the bottom of the trap formed in k parallel to space by the lower polariton branch are found to play a supplementary role among the processes governing P-thr.