We report on electron spin resonance (ESR) studies of the spin relaxation in Cs$_2$CuCl$_4$. The main source of the ESR linewidth at temperatures $T ≤ 150$ K is attributed to the uniform Dzyaloshinskii-Moriya interaction. The vector components of the Dzyaloshinskii-Moriya interaction are determined from the angular dependence of the ESR spectra using a high-temperature approximation. Both the angular and temperature dependence of the ESR linewidth have been analyzed using a self-consistent quantum-mechanical approach. In addition analytical expressions based on a quasi-classical picture for spin fluctuations are derived, which show good agreement with the quantum-approach for temperatures $T ≥ 2J/k_{\rm B} ≈ 15$ K. A small modulation of the ESR linewidth observed in the $ac$-plane is attributed to the anisotropic Zeeman interaction, which reflects the two magnetically nonequivalent Cu positions.