We present a comprehensive photoluminescence study of the exchange interaction in self-assembled CdTe/ZnTe quantum dots. We exploit the presence of multiple charge states in the photoluminescence spectra of single quantum dots to analyze simultaneously the fine structure of different excitonic transitions, including the recombination of neutral exciton and biexciton, doubly charged negative exciton, and negatively charged biexciton. We demonstrate that the fine structure results from the electron-hole exchange interaction and that a spin Hamiltonian with effective exchange constants can provide a good description of each transition in magnetic field for Faraday and Voigt field geometry. We determine and discuss values of the effective exchange constants for a large statistics of quantum dots.