We present a quantitative morphological analysis of 187 galaxies in a region covering the central 0.28 deg2 of the Coma Cluster. Structural parameters from the best-fitting Sérsic r1/n bulge plus, where appropriate, exponential disk model, are tabulated here. This sample is complete down to a magnitude of R=17 mag. By examining the recent compilation by Edwards et al. of galaxy redshifts in the direction of Coma, we find that 163 of the 187 galaxies are Coma Cluster members and that the rest are foreground and background objects. For the Coma Cluster members, we have studied differences in the structural and kinematic properties between early- and late-type galaxies and between the dwarf and giant galaxies. Analysis of the elliptical galaxies reveals correlations among the structural parameters similar to those previously found in the Virgo and Fornax Clusters. Comparing the structural properties of the Coma Cluster disk galaxies with disk galaxies in the field, we find evidence for an environmental dependence: the scale lengths of the disk galaxies in Coma are 30% smaller. An analysis of the kinematics shows marginal differences between the velocity distributions of elliptical galaxies with Sérsic index n<2 (dwarfs) and those with n>2 (giants), the dwarf galaxies having a greater (cluster) velocity dispersion. Finally, our analysis of all 421 background galaxies in the catalog of Edwards et al. reveals a nonuniform distribution in redshift with contrasts in density of ~3, characterized by a void extending from ~10,000 to ~20,000 km s-1, and two dense and extended structures centered at ~20,000 and ~47,000 km s-1.