Potassium (K) ion batteries offer a viable alternative to their lithium ion counterparts for large-scale energy storage applications due to lower production costs and comparably low standard redox potential. The choices of cathode and anode, the two battery components related to various parameters such as usable voltage range and capacity, are crucial for performance optimization. However, a fundamental understanding of phase formation in many cathode materials is yet to be established, which is important for realizing potential battery degradation mechanisms. By utilizing a statistical mechanics code and first principles Density Functional Theory (DFT) calculations, we systematically study ground state orderings of K_xCoO₂ with respect to K-ion composition x, where the cathode material is layered cobalt oxide (CoO₂). Our findings show that a multitude of phases form at different compositions of K-ions, some of which are the exact orderings found in a similar system, Na_xCoO₂. We also report the discovery of a new phase that is stabilized by undulating CoO₂ layers at high K-ion compositions, which has not been reported for smaller ions such as Li and Na in this cathode material. This is attributable to large alkali-alkali repulsions between K-ions, which suggests that synthesizing K_xCoO₂ at these compositions may lead to breakdown of the layered structure primarily due to strain energy effects. Accordingly, we develop a theoretical understanding of mechanical and electrical characteristics of K_xCoO₂ from first-principles for energy storage applications.