Officially, GACKT’s stage name is, well, GACKT, rendered in all caps and pronounced with a hard “a” as in “Jack”. Due to the structure of the Japanese language, it is generally pronounced as “Gakuto” in Japan. GACKT himself uses the Japanese pronunciation when speaking Japanese and the English pronunciation when speaking other languages.
His full name is, in Japanese 神威楽斗 (Kamui Gakuto), rendered as Gackt Camui in English.
His birth name is, despite what some people may tell you, unconfirmed (UPDATE: More recently confirmed as “Ooshiro Gakuto”). However, for reasons that will be obvious shortly, his stage name is unlikely to be his birth name.
The four kanji characters that make up “Kamui Gakuto” are 神(“god”, “divine”) 威(“might”) 楽(“music”, “enjoyment”) 斗(“to” sound, many meanings). 神威 shin’i is an actual Japanese word meaning “divine might”, however, here it is pronounced “kamui”, which is the word for a god in the language of the Ainu, a group native to northern Japan. Furthermore, in Ainu culture, all kamuy have names following a pattern such as Kotankor Kamuy, the god that protects (koru) the village (kotan). For actual humans, “Kamui” is not a common surname.
On the other hand “Gakuto” is a legitimate, though somewhat rare, male first name. It is generally written differently, using either 学 (learn) or 岳 (hill) for the “gaku” sound, with various kanji recruited for the ending “to” sound. 斗 is one of the characters often used for male names that end in “to”, and also in a lot of other situations where a “to” sound is needed, it has around 12 different dictionary meanings. Outside of names, it is most commonly used in the names of stars and constellations, and Japanese parents often pair it with other kanji connoting light or brightness. So it has a somewhat celestial feel.
Putting this altogether, 神威楽斗 approximately translates as “God of Music” or “the incarnation of divine power as music”. A slightly more tongue in cheek translation would be “God of Rock Stars”.