This paper presents the computation of the 5PM order contributions to the scattering angle and impulse of classical black hole scattering in the conservative sector at first self-force order (1SF) using the worldline quantum field theory (WQFT) formalism. The computation involves a four-loop calculation, which required advanced integration-by-parts and differential equation techniques on high-performance computing systems. The result is expressed in terms of multiple polylogarithms up to weight three, with no elliptic integrals appearing at 5PM order, unlike the 4PM order. The result was verified through internal consistency checks and external comparisons with the post-Newtonian (PN) literature and tail terms. Binary black hole (BH) and neutron star (NS) mergers are observed by gravitational wave detectors like LIGO-Virgo-KAGRA. Future detectors like LISA will improve experimental accuracy, enabling deeper insights into gravitational physics. Theoretical efforts aim to achieve high precision in gravitational waveforms from these events, combining perturbative and numerical approaches. The post-Newtonian (PN) and post-Minkowskian (PM) expansions are key tools in this effort, along with gravitational self-force (SF) methods. In the PM regime, the focus is on gravitational scattering of BHs or NSs. The compact bodies are modeled as point particles, and key observables like impulse and scattering angle are computed to high orders in the PM expansion. Spin and tidal effects have also been incorporated. Complementary perturbative QFT strategies have achieved comparable precision. The current state-of-the-art is 4PM order for the scattering angle and impulse. To advance to 5PM order, the spin-free four-loop contribution is needed. The SF expansion is a complementary perturbative scheme, assuming a hierarchy in masses. The PM loop expansion can be overlaid with the SF expansion, allowing for separate treatment of gauge-invariant SF sectors. The 5PM-1SF contribution to the impulse was computed using the WQFT formalism, with partial fraction identities used to planarize the integrand. The integration-by-parts (IBP) reduction was performed using an improved version of Kira. The result was verified through checks including momentum conservation, geodesic motion, and agreement with the PN literature. The function space for the scattering angle consists only of multiple polylogarithms (MPLs) up to weight three, with no elliptic functions. The scattering angle was computed using the impulse and verified against the PN literature. The result for the 5PM-1SF scattering angle is expressed as a sum of MPLs multiplied by coefficient polynomials. The result is compact and matches known results for the probe limit. The result is a significant step forward in understanding gravitational waveforms from binary systems.This paper presents the computation of the 5PM order contributions to the scattering angle and impulse of classical black hole scattering in the conservative sector at first self-force order (1SF) using the worldline quantum field theory (WQFT) formalism. The computation involves a four-loop calculation, which required advanced integration-by-parts and differential equation techniques on high-performance computing systems. The result is expressed in terms of multiple polylogarithms up to weight three, with no elliptic integrals appearing at 5PM order, unlike the 4PM order. The result was verified through internal consistency checks and external comparisons with the post-Newtonian (PN) literature and tail terms. Binary black hole (BH) and neutron star (NS) mergers are observed by gravitational wave detectors like LIGO-Virgo-KAGRA. Future detectors like LISA will improve experimental accuracy, enabling deeper insights into gravitational physics. Theoretical efforts aim to achieve high precision in gravitational waveforms from these events, combining perturbative and numerical approaches. The post-Newtonian (PN) and post-Minkowskian (PM) expansions are key tools in this effort, along with gravitational self-force (SF) methods. In the PM regime, the focus is on gravitational scattering of BHs or NSs. The compact bodies are modeled as point particles, and key observables like impulse and scattering angle are computed to high orders in the PM expansion. Spin and tidal effects have also been incorporated. Complementary perturbative QFT strategies have achieved comparable precision. The current state-of-the-art is 4PM order for the scattering angle and impulse. To advance to 5PM order, the spin-free four-loop contribution is needed. The SF expansion is a complementary perturbative scheme, assuming a hierarchy in masses. The PM loop expansion can be overlaid with the SF expansion, allowing for separate treatment of gauge-invariant SF sectors. The 5PM-1SF contribution to the impulse was computed using the WQFT formalism, with partial fraction identities used to planarize the integrand. The integration-by-parts (IBP) reduction was performed using an improved version of Kira. The result was verified through checks including momentum conservation, geodesic motion, and agreement with the PN literature. The function space for the scattering angle consists only of multiple polylogarithms (MPLs) up to weight three, with no elliptic functions. The scattering angle was computed using the impulse and verified against the PN literature. The result for the 5PM-1SF scattering angle is expressed as a sum of MPLs multiplied by coefficient polynomials. The result is compact and matches known results for the probe limit. The result is a significant step forward in understanding gravitational waveforms from binary systems.