Millimeter-wave (mmWave) communication is considered a promising technology for fifth-generation (5G) wireless communications systems since it can greatly improve system throughput. Unfortunately, because of extremely high frequency, mmWave transmission suffers from the signal blocking problem, which leads to the deterioration of transmission performance. In this paper, we solve this problem by the combination of ultra-dense network (UDN) and user-centric virtual cell architecture. The deployment of dense small base stations (SBSs) in UDN can reduce transmission distance of signals. The user-centric virtual cell architecture mitigates and exploits interference to improve throughput by using coordinated multipoint (CoMP) transmission technology. Nonetheless, the backhaul burden is heavy and interbeam interference still severe. Therefore, we propose a novel iterative backhaul capacity-limited joint user association and power allocation (JUAPA) scheme in ultra-dense mmWave networks under user-centric virtual cell architecture. To mitigate interference and satisfy quality of service (QoS) requirements of users, a nonconvex system throughput optimization problem is formulated. To solve this intractable optimization problem, we divide it into two alternating optimization subproblems, i.e., user association and power allocation. During each iteration, a many-to-many matching algorithm is designed to solve user association. Subsequently, we perform power allocation optimization using a successive convex approximation (SCA) algorithm. The results confirm that the performance of the proposed scheme is close to that of the exhaustive searching scheme, which greatly reduces complexity, and clearly superior to that of traditional schemes in improving system throughput and satisfying QoS requirements.