I know that Pluto and Charon have been found to circle round each other, hence the reason Pluto is no longer a planet I believe, which would mean that the centre of rotation is an arbitrary (!) point somewhere between the two of them. Where is it please? Can it be calculated?
It is listed in this wikipedia page.
Pluto's radius is listed there as 1,150 km; the radius of the barycenter is 2,110 km.
So the barycenter is 960 km above Pluto's surface!
(Note the radius of Pluto quoted differs to its main wikipedia page; that one says around 1,187 km)
I know that Pluto and Charon have been found to circle round each other, hence the reason Pluto is no longer a planet
This is not correct.
It is considered a dwarf planet because of a technical definition employed by the IAU in deciding what is a planet and is not. This technical definition relates to an object's ability to "sweep" it's orbit clear of asteroids and other small debris. The decision has nothing to do with Charon.
The centre of rotation is calculated using Newtonian physics. I think what you want is the Center Of Mass and related topics.
As StephenG stated, this is not the reason why Pluto is no longer a planet.
The center of rotation of the two bodies is called the barycenter. It can be calculated with accuracy, knowing the mass of the two bodies and the distance separating them. See :
The little ring around Pluto in the center describes its path. The barycenter of this binary system is on the opposite side of the ring, starting from Pluto.