We examine the structure and dynamics of the galaxies in the Coma cluster using a catalog of 552 redshifts including 243 new measurements. The velocity distribution is shown to be non-Gaussian due to structure associated with the group of galaxies around NGC4839, 40 arcmin SW of the cluster core. We apply a mixture-modelling algorithm to the galaxy sample and obtain a robust partition into two subclusters which we use to examine the system's dynamics. We find that the late-type galaxies are freely-falling into a largely virialised cluster core dominated by early types. We obtain a virial mass for the main cluster in close agreement with the estimates derived from recent X-ray data. The mass of the NGC4839 group is about 5-10% the mass of the main cluster. Assuming the main cluster and the NGC4839 group follow a linear two-body orbit, the favored solution has the two clusters lying at 74 degrees to the line of sight at a true separation of 0.8 Mpc and moving together at 1700 km/s. The cluster core shows evidence of an ongoing merger between two subclusters centered in projection on the dominant galaxies NGC4874 and NGC4889 but offset in velocity by 300 km/s and 1100 km/s respectively. Combining these results with X-ray and radio observations, and an interpretation of the presence or lack of an extended halo around the dominant galaxies, we develop a merger history for the Coma cluster.