In heterogeneous networks (HetNets), strong interference due to spectrum reuse affects each user's signal-to-interference ratio (SIR), and hence is one limiting factor of network performance. In this paper, we propose a user-centric interference nulling (IN) scheme in a downlink large-scale HetNet to improve coverage/outage probability by improving each user's SIR. This IN scheme utilizes at most maximum IN degree of freedom (DoF) at each macro-BS to avoid interference to uniformly selected macro (pico) users with signal-to-individual-interference ratio (SIIR) below a macro (pico) IN threshold, where the maximum IN DoF and the two IN thresholds are three design parameters. Using tools from stochastic geometry, we first obtain a tractable expression of the coverage (equivalently outage) probability. Then, we analyze the asymptotic coverage/outage probability in the low and high SIR threshold regimes. The analytical results indicate that the maximum IN DoF can affect the order gain of the outage probability in the low SIR threshold regime, but cannot affect the order gain of the coverage probability in the high SIR threshold regime. Moreover, we characterize the optimal maximum IN DoF which optimizes the asymptotic coverage/outage probability. The optimization results reveal that the IN scheme can linearly improve the outage probability in the low SIR threshold regime, but cannot improve the coverage probability in the high SIR threshold regime. Finally, numerical results show that the proposed scheme can achieve good gains in coverage/outage probability over a maximum ratio beamforming scheme and a user-centric almost blank subframes (ABS) scheme.