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The youngest of the brothers
The Nihon Jinmei Daijiten makes him the sixth son of Kuniie; the Nihon Kokugo Daijiten, the grandson of Kuniyori, whom tradition places at the head of the Awataguchi school. The latter gives him the title of Sakon Shōgen and the years Chōkan 1 to Kenchō 7 (1163-1255). Japanese Wikipedia gives those same dates with a question mark. No document of the time proves the kinship of the Awataguchi brothers, according to Markus Sesko. His elder brothers were, according to tradition, Kunitomo and Hisakuni, among others.
Smith of Go-Toba
He is said to have been a rotating smith (御番鍛冶, goban-kaji) of the retired emperor Go-Toba; the Nihon Kokugo Daijiten adds that he accompanied him into his exile on Oki. Sesko points out that the list of six rotating smiths of Oki is doubtful.
To Kamakura
According to the Kotō Meizukushi Taizen, he was born in Chōkan 1 (1163), moved to Yamanouchi, in Kamakura, at the age of 42, in Genkyū 2 (1205), at the request of the military government, and died in Kenchō 7 at the age of 93; the date of 1205 appears for the first time in the Nōami-bon Meizukushi, of 1483, according to Sesko. Sesko writes 1256 for Kenchō 7; Japanese Wikipedia and the Nihon Kokugo Daijiten give 1255.
Who summoned him is not clear. The Nōami-bon names the regent Hōjō Tokimasa (1138-1215) in 1205; almost all the other sources say Hōjō Tokiyori (1227-1263), during the Kenchō era (1249-1256). Sesko finds Tokiyori, a great connoisseur of swords, more plausible, and reckons that Kunitsuna would have worked in Kamakura for some six or seven years.
The beginning of sword-making in Sagami
In Kamakura he worked, with Ichimonji Sukezane and Saburō Kunimune, of Bizen, at the birth of local sword-making, according to Sesko. English Wikipedia records the tradition that Sōshū sword-making was born in 1249, when he forged a tachi for Hōjō Tokiyori, and that his son Shintōgo Kunimitsu is the first true Sōshū smith. Sesko presents this as the common theory, not as a fact.
The legend of the Onimaru
The Nihon Jinmei Daijiten tells the legend: the sword he forged for Tokiyori cut down the demon that tormented him. The old texts do not agree on the origin of the blade: the Kanchi-in-bon Meijin, of the late Kamakura, says that Hōjō Tokimune summoned an Awataguchi smith called 助綱 (Sukezuna), whom modern researchers take to be an error for Kunitsuna; the Ryūzōji-bon Meijin, of 1351, already says “Kunitsuna: Onimaru”, and the Utsunomiya-bon, of the early Muromachi, links it to Tokiyori, according to Japanese Wikipedia. Honma Kunzan saw in that Sukezuna an Ichimonji smith of Kamakura.