We present new measurements of the spatial distribution and kinematics of neutral hydrogen in the circumgalactic and intergalactic medium surrounding star-forming galaxies at z ∼ 2. Using the spectra of ≃ 3000 galaxies with redshifts 〈z〉 = 2.3 ± 0.4 from the Keck Baryonic Structure Survey, we assemble a sample of more than 200,000 distinct foreground-background pairs with projected angular separations of 3″ − 500″ and spectroscopic redshifts, with 〈zfg〉 = 2.23 and 〈zbg〉 = 2.57 (foreground, background redshifts, respectively.) The ensemble of sightlines and foreground galaxies is used to construct a 2-D map of the mean excess H i$\rm Ly\alpha$ optical depth relative to the intergalactic mean as a function of projected galactocentric distance ($20 {\,\, \lesssim \,\,}D_{\rm tran}/{\rm pkpc} {\,\, \lesssim \,\,}4000$) and line-of-sight velocity. We obtain accurate galaxy systemic redshifts, providing significant information on the line-of-sight kinematics of H i gas as a function of projected distance Dtran. We compare the map with cosmological zoom-in simulation, finding qualitative agreement between them. A simple two-component (accretion, outflow) analytical model generally reproduces the observed line-of-sight kinematics and projected spatial distribution of H i. The best-fitting model suggests that galaxy-scale outflows with initial velocity vout ≃ 600 km s−1 dominate the kinematics of circumgalactic H i out to Dtran ≃ 50 kpc, while H i at Dtran ≳ 100 kpc is dominated by infall with characteristic $v_{\rm in} {\,\, \lesssim \,\,}$ circular velocity. Over the impact parameter range $80 {\,\, \lesssim \,\,}D_{\rm tran}/{\rm pkpc} {\,\, \lesssim \,\,}200$, the H i line-of-sight velocity range reaches a minimum, with a corresponding flattening in the rest-frame $\rm Ly\alpha$ equivalent width. These observations can be naturally explained as the transition between outflow-dominated and accretion-dominated flows. Beyond Dtran ≃ 300 pkpc (∼1 cMpc), the line of sight kinematics are dominated by Hubble expansion.