Here I present a comprehensive study of the multiplicity of a volume-complete sample of K stars within 33 pc, a part RECONS K Star project (RKS), a large effort with the objective of searching for stellar, substellar, and planetary companions using three observational techniques to cover separations from 10 000 AU to 0.1 AU. In this work, I present the results of the companion search for 804 K dwarf type stars using the radial velocity (RV) technique. The survey extends for five years so far using the CHIRON Spectrograph at the CTIO/SMARTS 1.5m, achieving precisions down to 7 m/s for K dwarfs with V magnitudes between 7.0-11.5. Of the 804 K dwarfs within 33 pc and between DEC +30o and -30o, a sample of 562 K dwarfs did not have high precision RV measurements before, and are now the target of our volume-complete survey. Among the 804 stars we have found 124 RV perturbations consistent with companions never detected before, of which 38 are newly discovered stellar/sub-stellar orbits, 4 are likely planet candidates, and 82 are strong candidates for companions. Combining these results with known companions, we present here a detailed portrait of K dwarf systems unveiling a stellar multiplicity fraction of 22%, when only RV surveys are considered. The results remain consistent between the 25 pc and 33 pc, even when the 33 pc is almost doubling the stars in the 25 pc sample. The orbital architectures given by RVs and presented here, show that the lack of brown dwarfs in short period regimes still applies, and shows circular orbits become more rare with increasing orbital period. All of which starts to provide insights into formation processes around K stars. Ultimately, by providing a careful defined sample, a systematic search, and the combination of previous studies, this thesis research aims to set basis for understanding star and planet formation processes for decades to come