Chemotaxis is a phenomenon whereby organisms like ameba direct their movements responding to their environmental gradients, often called gradient climbing. It is considered to be the origin of self-movement that characterizes life forms. In this work, we have simulated the gradient climbing behaviour on Neural Cellular Automata (NCA) that has recently been proposed as a model to simulate morphogenesis. NCA is a cellular automata model using deep networks for its learnable update rule and it generates a target cell pattern from a single cell through local interactions among cells. Our model, Gradient Climbing Neural Cellular Automata (GCNCA), has an additional feature that enables itself to move a generated pattern by responding to a gradient injected into its cell states.