We have performed x-ray magnetic circular dichroism (XMCD) and valence-band photoemission studies of the diluted ferromagnetic semiconductor Zn$_{1-x}$Cr$_x$Te. XMCD signals due to ferromagnetism were observed at the Cr 2p absorption edge. Comparison with atomic multiplet calculations suggests that the magnetically active component of the Cr ion was divalent under the tetrahedral crystal field with tetragonal distortion along the crystalline a-, b-, and c-axes. In the valence-band spectra, spectral weight near the Fermi level was strongly suppressed, suggesting the importance of Jahn-Teller effect and the strong Coulomb interaction between the Cr 3d electrons.