Quality-Diversity (QD) algorithms have recently gained traction as optimisation methods due to their effectiveness at escaping local optima and capability of generating wide-ranging and high-performing solutions. Recently, Multi-Objective MAP-Elites (MOME) extended the QD paradigm to the multi-objective setting by maintaining a Pareto front in each cell of a MAP-ELITES grid. MOME achieved a global performance that competed with NSGA-11 and SPEA2, two well-established multi-objective evolutionary algorithms, while also acquiring a diverse repertoire of solutions. However, MOME is limited by non-directed genetic search mechanisms which struggle in high-dimensional search spaces. In this work, we present Multi-Objective MAP-Elites with Policy-Gradient Assistance and Crowding-based Exploration (MOME-PGX: a new QD algorithm that extends MOME to improve its data efficiency and performance. MOME-PGX uses gradient-based optimisation to efficiently drive solutions towards higher performance. It also introduces crowding-based mechanisms to create an improved exploration strategy and to encourage greater uniformity across Pareto fronts. We evaluate MOME-PGX in four simulated robot locomotion tasks and demonstrate that it converges faster and to a higher performance than all other baselines. We show that MOME-PGX is between 4.3 and 42 times more data-efficient than MOME and doubles the performance of MOME, NSGA-11 and SPEA2 in challenging environments.