Galaxy clusters are expected to be dark matter (DM) reservoirs and storage rooms for the cosmic-ray protons (CRp) that accumulate along the cluster's formation history. Accordingly, they are excellent targets to search for signals of DM annihilation and decay at gamma-ray energies and are predicted to be sources of large-scale gamma-ray emission due to hadronic interactions in the intracluster medium. We estimate the sensitivity of the Cherenkov Telescope Array (CTA) to detect diffuse gamma-ray emission from the Perseus galaxy cluster. We perform a detailed spatial and spectral modelling of the expected signal for the DM and the CRp components. For each, we compute the expected CTA sensitivity. The observing strategy of Perseus is also discussed. In the absence of a diffuse signal (non-detection), CTA should constrain the CRp to thermal energy ratio within the radius $R_{500}$ down to about $X_{500}<3\times 10^{-3}$, for a spatial CRp distribution that follows the thermal gas and a CRp spectral index $\alpha_{\rm CRp}=2.3$. Under the optimistic assumption of a pure hadronic origin of the Perseus radio mini-halo and depending on the assumed magnetic field profile, CTA should measure $\alpha_{\rm CRp}$ down to about $\Delta\alpha_{\rm CRp}\simeq 0.1$ and the CRp spatial distribution with 10% precision. Regarding DM, CTA should improve the current ground-based gamma-ray DM limits from clusters observations on the velocity-averaged annihilation cross-section by a factor of up to $\sim 5$, depending on the modelling of DM halo substructure. In the case of decay of DM particles, CTA will explore a new region of the parameter space, reaching models with $\tau_{\chi}>10^{27}$s for DM masses above 1 TeV. These constraints will provide unprecedented sensitivity to the physics of both CRp acceleration and transport at cluster scale and to TeV DM particle models, especially in the decay scenario.