1. The tautological answer to the question "what is the centroid $x_C$ of $C$?" is "the centroid $x_C$ of $C$ is the centroid $x_C$ of $C$". It is hardly possible to give a better answer to this question without having the terms in which to express $x_C$ specified.  

2. The answer to your second question, "How is it related to $x_A+x_B$?" is "in no reasonable way". 

For instance, it is easy to find two compact convex subsets $A$ and $B$ of $\mathbb{R}^n$ with $x_A=x_B=0$ but with $x_C=x_{A+B}$ being any prescribed vector in $\Bbb R^n$. 

Indeed, e.g. let $n=2$, let $A$ be the convex hull of the set $\{(-1,0),(1,0)\}$, and let $B$ be the convex hull of the set $\{(-2,0),(1,-1),(1,1)\}$. Then $x_A=x_B=(0,0)$ and 
$C=A+B$ is the convex hull of the set $\{(2,1),(2,-1),(-3,0),(0,1),(0,-1)\}$, so that $x_C=(1/7,0)\ne{0,0}=x_A+x_B$. 

Now, by rescaling, rotating, and embedding $\Bbb R^2$ into $\Bbb R^n$,  for any natural $n\ge2$ and any nonzero vector $v\in\Bbb R^n$ we get compact convex subsets $A$ and $B$ of $\mathbb{R}^n$ with $x_A=x_B=0$ but with $x_C=x_{A+B}=v$.