Query processing on cloud database systems is a challenging problem due to the dynamic cloud environment. The configuration and utilization of the distributed hardware used to process queries change continuously. A query optimizer aims to generate query execution plans (QEPs) that are optimal meet user requirements. In order to achieve such QEPs under dynamic environments, performing query re-optimizations during query execution has been proposed in the literature. In cloud database systems, besides query execution time, users also consider the monetary cost to be paid to the cloud provider for executing queries. Thus, such query re-optimizations are multi-objective optimizations which take both time and monetary costs into consideration. However, traditional re-optimization requires accurate cost estimations, and obtaining these estimations adds overhead to the system, and thus causes negative impacts on query performance. To fill this gap, in this paper, we introduce ReOptRL, a novel query processing algorithm based on deep reinforcement learning. It bootstraps a QEP generated by an existing query optimizer and dynamically changes the QEP during the query execution. It also keeps learning from incoming queries to build a more accurate optimization model. In this algorithm, the QEP of a query is adjusted based on the recent performance of the same query so that the algorithm does not rely on cost estimations. Our experiments show that the proposed algorithm performs better than existing query optimization algorithms in terms of query execution time and query execution monetary costs..