Complex sentence: The reason why Akitoki sought out the gang was because of his demonic blade, which belongs to his family, which he is on a quest to purify when he encounters Inuyasha and the others.
Split into simpler sentences: The reason why Akitoki sought out the gang was because of his demonic blade. The blade belongs to his family, which he is on a quest to purify when he encounters Inuyasha and the others.