The mystery of Undertaker is still open, so there are plenty of space for theories.
First let's gather the informations that we have already:
Undertaker is a grim reaper, now a deserter. In the past he commited suicide, so became a shinigami.
Undertaker's chain is an important clue. Many names stand on this chain, it includes the name of Cloudia Phantomhive.
The hair chain is his "treasure" and he trusted Fake Ciel (or Our Ciel) to take care of this treasure.
Undertaker and Vincent Phantomhive had a close relationship/friendship. The death of Vincent was painful for Undertaker.
He bring real Ciel back to life, because he didn't wanted to lose more Phantomhives.
Now I would like to share my theory with you.
I think that Undertaker was once member of the Phantomhive family. Maybe he married into the family. Undertaker's feelings and connection to the Phantomhives are so strong , that's why I think, only friendship cannot cover the truth.
Undertaker said many times he doesn't want to lose more Phantomhives. In my opinion, it's not just Vincent but Cloudia too. Or there are others too...
He lost his beloved ones and because of the sorrow he commited suicide. But then he became a grim reaper and decided to save Real Ciel from death.
The connection between Undertaker and Our Ciel is a strange one. They fought each other many times. I believe, that Undertaker didn't want to hurt Our Ciel. Undertaker hates rather Sebastian, because he will once take Ciel's life.
Undertaker cares about Our Ciel. He wants to protect him and separate him from Sebastian.
I know there is more behind everything, but there is still not enough clues.
To sum up, Undertaker is not that bad as he seems to be. He tries just desperately bring back life and escape from sorrow and pain.
[Thanks for reading! I would like to highlight, that this is just my humble opinion and speculation. Thank you guys for all the likes, reblogs and comments what I got so far!! ❤]