Galaxy clusters are expected to be both 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 γ-ray energies and are predicted to be sources of large-scale γ-ray emission due to hadronic interactions in the intracluster medium (ICM). In this paper, we estimate the sensitivity of the Cherenkov Telescope Array (CTA) to detect diffuse γ-ray emission from the Perseus galaxy cluster. We first perform a detailed spatial and spectral modelling of the expected signal for both the DM and the CRp components. For each case, we compute the expected CTA sensitivity accounting for the CTA instrument response functions. The CTA observing strategy of the Perseus cluster is also discussed. In the absence of a diffuse signal (non-detection), CTA should constrain the CRp to thermal energy ratio X₅₀₀ within the characteristic radius R₅₀₀ down to about X₅₀₀ < 3 × 10^-3, for a spatial CRp distribution that follows the thermal gas and a CRp spectral index α_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 α_CRp down to about Δα_CRp ≃ 0.1 and the CRp spatial distribution with 10% precision, respectively. Regarding DM, CTA should improve the current ground-based γ-ray DM limits from clusters observations on the velocity-averaged annihilation cross-section by a factor of up to ∼ 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 τ_χ > 10²⁷ 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.