We report a systematic first-principles study on the recent discovered superconducting Ba$_{1-x}$K$_x$Fe$_2$As$_2$ systems ($x$ = 0.00, 0.25, 0.50, 0.75, and 1.00). Previous theoretical studies strongly overestimated the magnetic moment on Fe of the parent compound BaFe$_2$As$_2$. Using a negative on-site energy $U$, we obtain a magnetic moment 0.83 $mu_B$ per Fe, which agrees well with the experimental value (0.87 $mu_B$). K doping tends to increase the density of states at fermi level. The magnetic instability is enhanced with light doping, and is then weaken by increasing the doping level. The energetics for the different K doping sites are also discussed.
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