With the advantages of lightweight and high resource utilization, cloud-native technology with containers as the core is gradually becoming the mainstream technical architecture for information infrastructure. However, malware attacks such as Doki and Symbiote threaten the container runtime's security. Malware initiates various types of runtime anomalies based on process form (e.g., modifying the process of a container, and opening the external ports). Fortunately, dynamic monitoring mechanisms have proven to be a feasible solution for verifying the trusted state of containers at runtime. Nevertheless, the current routine dynamic monitoring mechanisms for baseline data protection are still based on strong security assumptions. As a result, the existing dynamic monitoring mechanism is still not practical enough. To ensure the trustworthiness of the baseline value data and, simultaneously, to achieve the integrity verification of the monitored process, we combine blockchain and trusted computing to propose a process integrity monitoring system named IPMS. Firstly, the hardware TPM 2.0 module is applied to construct a trusted security foundation for the integrity of the process code segment due to its tamper-proof feature. Then, design a new format for storing measurement logs, easily distinguishing files with the same name in different containers from log information. Meanwhile, the baseline value data is stored on the blockchain to avoid malicious damage. Finally, trusted computing technology is used to perform fine-grained integrity measurement and remote attestation of processes in a container, detect abnormal containers in time and control them. We have implemented a prototype system and performed extensive simulation experiments to test and analyze the functionality and performance of the PIMS. Experimental results show that PIMS can accurately and efficiently detect tampered processes with only 3.57% performance loss to the container.