If Pluto is tidally locked with Charon, is its rotation still slowing down? I know that a a planet or moon's rotation will slow down or speed up until it is tidally locked with a body, is Pluto's rotation being affected by the sun or is it being forced by Charon just to to be tidally locked with Charon?
Short answer.  Pluto's rotation is 100% governed by Charon, or, more accurately, the Pluto-Charon system.  The sun is irrelevant to it's rotation.

Small point to add.  Pluto's rotation is pretty much governed by the Pluto-Charon orbit.  If Charon moves closer to Pluto, Pluto's rotation will correspondingly increase.   If Charon moves further away, Pluto's rotation will decrease.  That's how tidal locking works.   I answered the question in regards to the orbit changing, not the rotation changing, but in a tidally locked system the rotation period follows the orbital period.

Long answer:


  I know that a a planet or moon's rotation will slow down or speed up
  until it is tidally locked with a body,


Tidal locking doesn't always happen.  For distant orbits the tidal force that makes tidal locking happen is often very weak and very slow, often requiring more time than the lifespan of the star in that solar system.

Also, for high eccentricity orbits, classic tidal locking is impossible though resonant orbits like Mercury's 3:2 around the sun are probably fairly common.   And many moons will escape their planet or crash into their planet long before mutual tidal locking occurs, though it's not uncommon for moons themselves to be tidally locked.   Also small moons don't have sufficient gravity to pull the object they orbit into mutual tidal locking.  

For mutual tidal locking, like Pluto-Charon, there needs to be respectible mass ratio or, perhaps an initial formation where orbital speed and rotation are very close, so it doesn't take much force to lead to mutual tidal locking.  Our moon, at 1/81st the mass of the Earth, would eventually cause the Earth to tidal lock to it, but estimates say it will take longer than it will take the sun to go red-giant for that to happen, and the expected outcome of our moon is that it will crash into the Earth before there is mutual tidal locking.  

So, mutual tidal locking like pluto-charon is actually pretty rare because you need the smaller object to have a significant percentage of mass to the larger one.   One-way tidal locking is quite common, but your question addresses mutual tidal locking.


  is Pluto's rotation being affected by the sun or is it being forced by
  Charon just to to be tidally locked with Charon?


For a quick approximation, the tidal force can roughly/bad approximation be estimated by the size of the object in the sky times it's density.   There's other factors, ofcourse, like the size of the object receiving the tidal force, but for a rough approximation, the size in the sky gives an estimate.   For example, the sun and moon each take up about 1/2 a degree of arc when viewed from the surface of the Earth, so they have similar tidal forces due to having similar size.   The major difference being that the Moon is about 2.4 times more dense than the sun so it's tides are correspondingly about 2.4 times as strong.

This visual size approximation doesn't work for very-near-earth orbits, as it's the size that would appear from the center of the Earth, not the surface that should be applied.

Charon from the surface of Pluto is huge.   It spans several degrees of arc, over 10 times the size the Moon appears from the Earth.   The sun, from Pluto is tiny about 1/30th the size it appears from Earth, which is only about 50% larger than Venus appears in the sky at it's largest, or, about 1 minute of arc, but even at that distance the sun is still bright enough to cast visible shadows and would probably damage your retina if you stared at it.   It would be the most impressive light in the sky from Pluto, but still tiny, so the tidal effect on Pluto-Charon would be quite small.  (I'd prefer not to do the math, my math tends to run into errors and I'm not good with mathjax).

But the sun's tidal influence on the Pluto-Charon system is close to negligible, It can be calculated and it's not zero, but it's very small.   Over a very very very long time, the Sun's tidal force will sap energy from the Pluto-Charon system and draw them closer towards each other, but at that distance, billions maybe trillions of years.  Pluto is probably more likely to meet a different fate well before then, such as crashing into Neptune or being thrown out of the solar-system before the Pluto-Charon system becomes unstable.  

It would be different if Pluto-Charon orbited at Mercury or Venus distance (ignoring the obvious melting of ices and outgassing in a warmer environment).  That close, the sun becomes a major player in the future of the mutually tidally-locked systems, likely causing them to crash into each other, but at Pluto distance, there's virtually no effect.



When a system is mutually tidally locked with a nearly circular orbit, then it's very stable.  If there's no tidal locking or one way tidal locking like Earth-Moon, then there's a tidal bulge that leads or follows the moon and that creates a force that can slow down or speed up the planet and push the moon away or draw it in closer.   With tidal locking, there's no leading or following with the tidal bulge, so the orbit is less prone to change.   There's always some orbital variation and some orbital energy converted to heat due to no orbits being perfectly circular, but mutual tidal locking is still a very stable orbital system with less change than most, as long as there isn't a 3rd party forcing change, such as a debris cloud creating drag or outgassing or orbiting close to a massive body that saps energy from the tidally locked orbit. 

No orbits are stable forever, and considering all the factors, Pluto and Charon are close enough that they exchange some atmosphere and as the sun grows more luminous over time, that exchange will probably increase, and likely both are losing mass slowly over time due to the solar wind.    This very gradual loss of mass, I would think, would cause the orbit to expand, but very slowly.   With no loss of gas by the solar wind, the Pluto-Charon orbit should very slowly contract due to tidal forces from the sun and some orbital energy being converted into heat, but both of those effects are very slow.   Mutual tidal locking is for the most part a very stable system.

Hope that wasn't too long, but that's the gist of it.   My answer could probably be improved with some math and some linked references but that would make it even longer.