This paper considers the detection of fluctuating target in heavy-tailed clutter through the use of dynamic programming based on track-before-detect (DP-TBD) in radar systems. The clutter is modeled in terms of K-distribution, which can be widely used to describe non-Gaussian clutter received from high-resolution radars and radars working at small grazing angle. Swerling type 1 is considered to describe the target fluctuation between scans. Conventional TBD techniques suffer from significant performance loss in heavy-tailed environments due to the more frequent occurrences of target-like outliers. In this paper, we resort to DP-TBD algorithm based on prior information, which can enhance the detection performance by using the environment and target fluctuating information during the integration process of TBD. Under non-Gaussian background, the expressions of the likelihood ratio merit function for Swerling type 1 target are derived first. However, the closed analytical form of the merit function is difficult to be obtained. In order to reduce the complexity of evaluating the merit function and the computational load, an efficient approximation method as well as a two-stage detection approach is proposed and used in the integration process. Finally, several numerical simulations of the new strategy and the comparisons are presented to verify that the proposed algorithm can improve the detection performance, especially for fluctuating target in heavy-tailed clutter.