Thermally-activated 1/2〈111〉 screw dislocation motion is the controlling plastic mechanism at low temperatures in body-centered cubic (bcc) crystals. Dislocation motion proceeds by nucleation and propagation of atomic-sized kink pairs in close-packed planes. The atomistic character of kink pairs can be studied using techniques such as molecular dynamics (MD). However, MD's natural inability to properly sample thermally-activated processes as well as to capture {110} screw dislocation glide calls for the development of other methods capable of overcoming these limitations. Here we develop a kinetic Monte Carlo (kMC) approach to study single screw dislocation dynamics from room temperature to 0.5Tm and at stresses 0<σ<0.9σP, where Tm and σP are the melting point and the Peierls stress. The method is entirely parameterized with atomistic simulations using an embedded atom potential for tungsten. To increase the physical fidelity of our simulations, we calculate the deviations from Schmid's law prescribed by the interatomic potential used and we study single dislocation kinetics using both projections. We calculate dislocation velocities as a function of stress, temperature, and dislocation line length. We find that considering non-Schmid effects has a strong influence on both the magnitude of the velocities and the trajectories followed by the dislocation. We finish by condensing all the calculated data into effective stress and temperature dependent mobilities to be used in more homogenized numerical methods.