Is it correct for a PhD holder to sign as "Dr. J. Doe, PhD"? I have seen some of the PhD holders sign as "Dr. [First Name] [Surname], PhD".

Is it academically correct to write "Dr. Frank Amoani Arthur, PhD"?
There are two important points to note about name markers that refer to academic titles:


You can have more than one of these markers in your name: Kay Doe, PhD, MD indicates a person who is both a doctor of philosophy and a doctor of medicine.
The marker Dr. can very often be used as a variant of the post-positioned markers. If both of them are equivalent markers in your country, you can choose either marker to refer to your academic title.


These two observations make the suggested name form Dr. Kay Doe, PhD rather ambiguous. 

Does it refer to a person with two doctor titles because there are two markers? Or do both markers refer to just one doctor title? But if so, why is that doctor referred to twice in one name? This may even lead to suspicion: Does the name bearer use two markers which refer to the same  title with the intention to sound more impressive than their academic distinction actually warrants?

This ambiguity is avoided if you use only one title marker per title.