The motion of a deformable active particle in linear shear flow is explored theoretically. Based on symmetry considerations, we propose coupled nonlinear dynamical equations for the particle position, velocity, deformation, and rotation. In our model, both, passive rotations induced by the shear flow as well as active spinning motions, are taken into account. Our equations reduce to known models in the two limits of vanishing shear flow and vanishing particle deformability. For varied shear rate and particle propulsion speed, we solve the equations numerically in two spatial dimensions and obtain a manifold of different dynamical modes including active straight motion, periodic motions, motions on undulated cycloids, winding motions, as well as quasi-periodic and chaotic motions induced at high shear rates. The types of motion are distinguished by different characteristics in the real-space trajectories and in the dynamical behavior of the particle orientation and its deformation. Our predictions can be verified in experiments on self-propelled droplets exposed to a linear shear flow.