This is merely a couple of examples showing how $HH_*(\mathbf{C}_A)$ may behave.
Let first $A$ be the polynomial algebra $\Bbb C[x]$. Then $\mathbf{C}_A$ is the
category of coherent sheaves with zero-dimensional support on $\Bbb A^1$. Since sheaves with supports on different points are orthogonal,this category is just the direct sum over all closed points of $\Bbb A^1$ of the corresponding categories, $\mathbf{C}_A = \oplus_{\lambda \in \Bbb C}~ \mathrm{Coh}_{\lambda}(\Bbb A^1).$
$Coh_\lambda(\Bbb A^1)$ is the category of vector spaces endowed with an endomorphism with all eigenvalues equal to $\lambda$. In particular, all these categories are equivalent to each other and, in particular, to $\mathrm{Coh}_{0}(\Bbb A^1)$, the category of
vector spaces with nilpotent endomorphism, in other words, to
$\mathbf{C}_{\Bbb C[[x]]}$, where $\Bbb C[[x]]$ is the power series algebra. Now, there's a localization sequence of
triangulated categories:
$$\mathbf{C}_{\Bbb C[[x]]} \longrightarrow \Bbb C[[x]]-\mathrm{mod}_{\mathrm{fin. gen.}}
\longrightarrow
\Bbb C[x^{-1}, x]] - \mathrm{vect}_{\mathrm{fin. dim.}}$$
where $\Bbb C[x^{-1}, x]] - \mathrm{vect}_{\mathrm{fin. dim.}}$ is the category of finite dimensional vector spaces over the field of Laurent series. This sequence gives you a long exact sequence of Hochschild homology
$$\dots \longrightarrow HH_0(\mathbf{C}_{\Bbb C[[x]]})
\longrightarrow
HH_0(\Bbb C[[x]]-\mathrm{mod})
\longrightarrow
HH_0(\Bbb C[x^{-1}, x]] - \mathrm{vect})
\longrightarrow
HH_{-1}(\mathbf{C}_{\Bbb C[[x]]})
\longrightarrow 0.$$
(see Keller, "On the cyclic homology of exact categories", https://www.sciencedirect.com/science/article/pii/S0022404997001527)
In particular, you can see that $HH_{-1}(\mathbf{C}_{\Bbb C[[x]]})$ is non-zero and is isomorphic to the polynomials in $x^{-1}$.
I don't know for sure, but I'd think that using something like adelic resolutions will show you that the category of sheaves with zero-dimensional support on a smooth variety of dim $n$ will have non-zero (in general) $HH_k$ for $k > -n$.
On the other hand, if you take $U{\mathfrak g}$, the universal enveloping algebra of
some semisimple Lie algebra, then $\mathbf{C}_{U{\mathfrak g}}$ will be semisimple, so only $HH_0(\mathbf{C}_{U{\mathfrak g}})$ will be non-zero,
while $HH_*(U\mathfrak g -\mathrm{mod})$ is isomorphic to the Lie algebra
homology of $\mathfrak g$ with coefficients in the ring of invariant polynomials and is non-zero in arbitrary large degrees.
And of course if your algebra was finite dimensional from the beginning, you'll get nothing new.
