Trying to understand the chemistry in an iron carbon battery
Ions in solution


  [OP] I understand that ferrous chloride dissociates in water to $\ce{Fe^2+}$ and $\ce{Cl-}$ ions.


Yes. The same goes for the iron(III) species.


  [OP] I'm told that iron(II) chloride loses electrons to the positive electrode and becomes iron(III) chloride.


No, iron(II) loses electrons and becomes iron(III). The chloride is less relevant, and could be exchanged for a different ion.


  [OP]  How does this happen if the iron(II) chloride has dissociated in to ions, there is no iron(II) chloride anymore?


The reactant is $\ce{Fe^2+(aq)}$, not $\ce{FeCl2(s)}$.

Ion movement


  [OP] During charge the $\ce{Fe^2+}$ ions are attracted to to the negative iron electrode and the $\ce{Cl-}$ ions are attracted to the positive carbon electrode. 


Yes, but a small fraction, only. Otherwise the repulsion between like ions would be too great.


  [OP] My problem is understanding what happens at the positive carbon electrode. I'm told that iron(II) chloride loses electrons to the positive electrode and becomes iron(III) chloride. 


Just $\ce{Fe^3+(aq)}$. There might be a double layer, but no solid iron(III) chloride.


  [OP] Fine, but how does this happen if all the $\ce{Fe^2+}$ ions are at the negative electrode?


There are plenty of cations in the bulk of the solution, and plenty near either electrode.