It is known that Arima Kishou was the One Eyed King. He had taken the title because even though he was a half-human (physically human but with the abilities/senses of a ghoul, excluding the kagune). … And so it was spread that the one who kills the One Eyed King will be the hope and saviour for all ghouls.
Kaneki turn into a Kakuja because of repeated cannibalism (higher ghoul to human ratio). It is said that through cannibalism, a ghoul absorbs RC cells, and this increases the concentration of RC cells within a ghoul’s body. A Kakuja can form through continuous acts of cannibalism or extreme levels of stress and trauma.
Kaneki’s time at the CCG as Haise gave him practice and training under Arima – the white reaper of CCG. That made him better with sword art and earned him the name ‘black reaper’. His emotional side is what gives him the most strength. That side of him gave him strength to kill Yamori and take on Arima.
The main reason as to why Kaneki is so strong is because he is a half ghoul and not a full ghoul. It is mentioned that half ghouls are more powerful than normal ghouls. This can be proven as Ken and Eto (or the One-Eyed Owl) are the two most powerful ghouls in the anime and both of them are half ghouls.