We present a Greens function approach to calculate the Dzyaloshinskii-Moriya interactions (DMI) from first principles electronic structure calculations, that is computationally more efficient and accurate than the most-commonly employed supercell and generalized Bloch-based approaches. The method is applied to the (111) Co/Pt bilayer where the Co- and/or Pt-thickness dependence of the DMI coefficients are calculated. Overall, the calculated DMI are in relatively good agreement with the corresponding values reported experimentally. Furthermore, we investigate the effect of strain in the DMI tensor elements and show that the isotropic N{e}el DMI can be significantly modulated by the normal strains, $epsilon_{xx},epsilon_{yy}$ and is relatively insensitive to the shear strain, $epsilon_{xy}$. Moreover, we show that anisotropic strains, $(epsilon_{xx}-epsilon_{yy})$ and $epsilon_{xy}$, result in the emergence of anisotropic N{e}el- and Bloch-type DMIs, respectively.
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