In this Letter, a high-power, high-brightness all-fiberized Raman amplifier based on a cladding-pumping scheme is presented for the first time, to the best of our knowledge. The triple-clad passive fiber is employed as Raman gain fiber in the laser system. The maximum output power is 762.6 W emitting at 1130 nm. To the best of our knowledge, this is the highest power in the fields of cladding-pumped Raman amplifiers. Through a cladding-pumping process, the beam quality parameter ${{\rm M}^2}$M2 improves from 6.12 of seed laser to 2.24 at maximum output power of 762.6 W, while the best ${{\rm M}^2}$M2 is 1.9 at 267.2 W. It is also the best beam quality of Raman laser with brightness enhancement in any kind of configuration (graded-index fiber or multi-clad fiber, laser or amplifier, all-fiber or free-space configuration) with power of over 100 W.