We develop a theoretical model for the thermodynamics and kinetics of clathrin self-assembly. Our model addresses the behavior in two dimensions and can be easily extended to three dimensions, facilitating the study of membrane, surface, and bulk assembly. The clathrin triskelia are modeled as flexible pinwheels that form leg-leg associations and resist bending and stretching deformations. Thus, the pinwheels are capable of forming a range of ring structures, including 5-, 6-, and 7-member rings that are observed experimentally. Our theoretical model employs Brownian dynamics to track the motion of clathrin pinwheels at sufficiently long time scales to achieve complete assembly. Invoking theories of dislocation-mediated melting in two dimensions, we discuss the phase behavior for clathrin self-assembly as predicted by our theoretical model. We demonstrate that the generation of 57 defects in an otherwise perfect honeycomb lattice resembles creation of two dislocations with equal and opposite Burgers vectors. We use orientational- and translational-order correlation functions to predict the crystalline-hexatic and hexatic-liquid phase transitions in clathrin lattices. These results illustrate the pivotal role that molecular elasticity plays in the physical behavior of self-assembling and self-healing materials. Show Partial Abstract