Yes, you can use chained possessives, although your examples are not quite correct.
Father of John's pen is an antique."
We would never refer to someone as "father of John". It would either need an article to be at least grammatically correct, but even then it would not be idiomatic. Using a chained possessive, we would say:
John's father's pen is an antique.
Dio's voice actor is also Shirogane's father's.
This isn't quite as bad, it feels a little ambiguous, but I suppose that is because I don't know the characters it refers to. In context, it might be ok, but if you don't know that Dio is a character it sounds like Dio owns the voice actor in some way. I would prefer:
Dio's voice actor also voices Shirogane's father.
This is also not a chained possessive - just a sentence that contains two possessives.
To avoid the ambiguity I mentioned you may have to dispense with the first possessive:
The actor that voices Dio also voices Shirogane's father.