Payment channels serve as an effective solution to the scalability problem of cryptocurrencies, which significantly increase transaction rates by allowing users to conduct large-scale offline transactions off-chain without posting everything to the blockchain. However, the existing payment channels lack privacy protection for the transaction amount and the linking relationship between the two parties to the transaction. Therefore, in order to address the scalability and privacy issues of cryptocurrencies such as Bitcoin, this paper proposes a zk-SNARKs-based anonymous payment channel (zk-APC), which supports an unlimited number of off-chain payments between the payer and the payee and protects the privacy of the participants. Specifically, the proposed scheme achieves relational anonymity and amount privacy for both on-chain and off-chain transactions in the payment channel through utilizing zero-knowledge proof (zk-SNARKs) and commitment schemes. This paper proves that the proposed method is more effective than similar schemes through a performance evaluation.