This paper investigates the slice-based service function chain embedding (SBSFCE) problem, which is to embed the service function chains (SFCs) of flows from different slices on a physical network for end-to-end network slice deployment. Compared with regarding slice deployment as complete virtual network embedding (VNE), deploying slices from the perspective of SBSFCE is beneficial for achieving more delicate resource allocation and jointly optimizing virtual network function (VNF) mapping and link mapping without the need for particular virtual topology designs. However, performing effective SBSFCE also faces several key challenges like diversified and differentiated requirements of flows, inter-slice and intra-slice VNF sharing, priority-aware admission control, and VNF placement restrictions, and few existing SBSFCE works have comprehensively considered or solved these challenges. In view of this, we address the SBSFCE problem by jointly considering the above key challenges in this paper. Specifically, we formulate the SBSFCE problem as an integer linear programming (ILP) that aims to maximize flow acceptance ratios and minimize network resource costs. Then, we propose two novel heuristic algorithms, weight-oriented embedding (WOE) and weight-oriented ratio embedding (WORE), to solve the problem. Simulation results demonstrate that our algorithms outperform benchmark algorithms and achieve near-optimal performance.