We provide a parton construction of wavefunctions and effective field theories for fractional Chern insulators. We also analyze a strong coupling expansion in lattice gauge theory that enables us to reliably map the parton gauge theory onto the microsopic Hamiltonian. We show that this strong coupling expansion is useful because of a special hierarchy of energy scales in fractional quantum Hall physics. Our procedure is illustrated using the Hofstadter model and then applied to bosons at 1/2 filling and fermions at 1/3 filling in a checkerboard lattice model recently studied numerically. Because our construction provides a more or less unique mapping from microscopic model to effective parton description, we obtain wavefunctions in the same phase as the observed fractional Chern insulators without tuning any continuous parameters.