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In this work structural refinement, complex impedance spectroscopy, and Raman spectroscopy have been investigated on Na$_{0.5}$(Bi$_{1-x}$Dy$_x$)$_{0.5}$TiO$_3$ (xDyNBT) ceramic systems. The pure NBT, 2DyNBT and 5DyNBT compounds crystallize in a rhombohedral structure while the 15DyNBT composition crystallizes in an orthorhombic Pnma structure. We reported that dysprosium addition affects the phase transition temperatures as well as the dielectric losses. The electrical transport at high temperatures was investigated using the CIS over a wide frequency range. The studied samples showed a non-Debye type process, with a short-range relaxation for the pure NBT and a coexistence of both localized and long-range relaxations of charge carriers for the 2DyNBT and 5DyNBT compounds. For the high concentration, 15DyNBT, a short-range relaxation is observed. Moreover, using a brick-layer model we discuss the resistance and capacitance of the different contributors (grain and grain boundaries) in our samples. High temperature Raman spectroscopy investigation was performed in order to follow the temperature evolution of the structural transformations on ferroelectric compounds. Anomalies in the temperature evolution of the vibrational modes are seen to correlate well with the temperature transitions observed from dielectric measurements.