We present the KMOS-CLASH (K-CLASH) survey, a K-band Multi-Object Spectrograph (KMOS) survey, of the spatially resolved gas properties and kinematics of 191 (pre-dominantly blue) H{alpha}-detected galaxies at 0.2~<z~<0.6 in field and cluster environments. K-CLASH targets galaxies in four Cluster Lensing And Supernova survey with Hubble (CLASH) fields in the KMOS IZ-band, over 7arcmin radius (~2-3Mpc) fields of view. K-CLASH aims to study the transition of star-forming galaxies from turbulent, highly star-forming disc-like and peculiar systems at z~1-3, to the comparatively quiescent, ordered late-type galaxies at z~0, and to examine the role of clusters in the build-up of the red sequence since z~1. In this paper, we describe the K-CLASH survey, present the sample, and provide an overview of the K-CLASH galaxy properties. We demonstrate that our sample comprises star-forming galaxies typical of their stellar masses and epochs, residing both in field and cluster environments. We conclude K-CLASH provides an ideal sample to bridge the gap between existing large integral-field spectroscopy surveys at higher and lower redshifts. We find that star-forming K-CLASH cluster galaxies at intermediate redshifts have systematically lower stellar masses than their star-forming counterparts in the field, hinting at possible 'downsizing' scenarios of galaxy growth in clusters at these epochs. We measure no difference between the star formation rates of H{alpha}-detected, star-forming galaxies in either environment after accounting for stellar mass, suggesting that cluster quenching occurs very rapidly during the epochs probed by K-CLASH, or that star-forming K-CLASH galaxies in clusters have only recently arrived there, with insufficient time elapsed for quenching to have occurred.