We have developed an atomistic kinetic Monte Carlo (kMC) model describing carbon diffusion, trapping and detrapping in the stress field generated by a 1/2 screw dislocation in bcc iron. By performing long time scale carbon diffusion simulations we study the formation of carbon Cottrell atmospheres around screw dislocations at different temperatures and background carbon concentrations. The kMC simulations allow us to predict the rate of formation and strength of carbon atmospheres which control the dislocation's behavior resulting in dynamic strain aging in steel. From the kMC simulations of carbon diffusivity in a block of iron crystal containing a screw dislocation we extract the effective diffusion constants. Our results show that except at low carbon concentrations the diffusivity resulting from carbon transport inside the dislocation core dominates the effective diffusivity, which indicates that pipe diffusion in a screw dislocation is likely. The kMC approach, which explicitly accounts for the behavior of individual carbon atoms, offers an atomistic view of the carbon drag mechanism by which mobile dislocations can collect and transport carbon within their cores. We simulate the carbon drag mechanism at different temperatures, background carbon concentrations, and dislocation velocities and estimate the maximal dislocation velocity at which the atmosphere of carbon atoms can follow a moving screw dislocation.