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We present an analytical study of a strongly correlated quantum dot-based thermoelectric particle-exchange heat engine for both finite and infinite on-dot Coulomb interaction. Employing Keldysh's non-equilibrium Green's function formalism for different decoupling schemes in the equation of motion, we have analyzed the thermoelectric properties within the non-linear transport regime. As the simplest mean-field approximation is insufficient for analyzing thermoelectric properties in the Coulomb blockade regime, one needs to employ a higher-order approximation to study strongly correlated QD-based heat engines. Therefore initially, we have used the Hubbard-\Romannum{1} approximation to study the quantum dot level position ($ε_d$), thermal gradient ($ΔT$), and on-dot Coulomb interaction ($U$) dependence of the thermoelectric properties. Furthermore, as a natural extension, we have used an approximation beyond Hubbard-\Romannum{1} in the infinite-$U$ limit (strong on-dot Coulomb repulsion) to provide additional insight into the operation of a more practical quantum dot heat engine. Within this infinite-$U$ limit, we examine the role of the symmetric dot-reservoir tunneling ($Γ$) and external serial load resistance ($R$) in optimizing the performance of the strongly correlated quantum dot heat engine. Our infinite-$U$ results show a good quantitative agreement with recent experimental data for a quantum dot coupled to two metallic reservoirs.