The Eyeball Man is a Vitreous Drinker employed by the Lady in White. From dialogue in the Episode 28 epilogue he appears to be a species, rather than a specific individual, with Swayne promising to get the LiW "another one of ... those" after it is killed trying to contain the Rakshasa mass-divination attack.
Little can be seen under the Eyeball Man's signature grey cloak, under which it is "unnaturally dark", suggesting some sort of magical darkness. What is apparent though, is a lot of eyeballs. This, coupled with his role in Swayne's information warfare division, implies that he is some sort of diviner.