Nonlinear optical responses are a crucial probe of physical systems including periodic solids. In the absence of electron-electron interactions, they are calculable with standard perturbation theory starting from the band structure of Bloch electrons, but the resulting formulas are often large and unwieldy, involving many energy denominators from intermediate states. This work gives a Feynman diagram approach to calculating nonlinear responses. This diagrammatic method is a systematic way to perform perturbation theory, which often offers shorter derivations and also provides a natural interpretation of nonlinear responses in terms of physical processes. Applying this method to second-order responses concisely reproduces formulas for the second-order-harmonic shift current. We then apply this method to third-order responses and derive formulas for third-order-harmonic generation and self-focusing of light, which can be directly applied to tight-binding models. Third-order responses in the semiclasscial regime include a Berry curvature quadrupole term, whose importance is discussed including symmetry considerations and when the Berry curvature quadrupole becomes the leading contribution. The method is applied to compute third-order optical responses for a model Weyl semimetal, where we find a new topological contribution that diverges in a clean material, as well as resonances with a peculiar linear character.