We present adaptive optics assisted integral field spectroscopy of nine Hα-selected galaxies at z = 0.84-2.23 drawn from the HiZELS narrowband survey. Our observations map the kinematics of these star-forming galaxies on kpc scales. We demonstrate that within the interstellar medium of these galaxies, the velocity dispersion of the star-forming gas (σ) follows a scaling relation σΣ^1/n _SFR + constant (where Σ_SFR is the star formation surface density and the constant includes the stellar surface density). Assuming the disks are marginally stable (Toomre Q = 1), this follows from the Kennicutt-Schmidt relation (Σ_SFR = AΣⁿ _gas), and we derive best-fit parameters of n = 1.34 ± 0.15 and A = 3.4^+2.5 _-1.6 × 10^-4 M _yr ^-1 kpc ^-2, consistent with the local relation, and implying cold molecular gas masses of M _gas = 10^9-10 _M and molecular gas fractions of M _gas(_Mgas + _M) = 0.3 ± 0.1, with a range of 10%-75%. We also identify 11 kpc-scale star-forming regions (clumps) within our sample and show that their sizes are comparable to the wavelength of the fastest growing mode. The luminosities and velocity dispersions of these clumps follow the same scaling relations as local H II regions, although their star formation densities are a factor 15 ± 5 × higher than typically found locally. We discuss how the clump properties are related to the disk, and show that their high masses and luminosities are a consequence of the high disk surface density.