As was already mentioned, the answer to the question asked is "no, we currently do not have a good classification". Here I wish to describe a successful and interesting recent line of research which does not aim at giving such a classification, rather merely at counting how many representations we do have.

To put things in context, let me first make the following observations:

the set of irreducible representations of a product of groups is in bijection with the product of the sets of irreducible representations of the groups.

$\text{SL}_n$ over a product of rings (commutative with 1) is isomorphic to the product of $\text{SL}_n$ over the rings.

for a natural $m$, $\mathbb{Z}/m\mathbb{Z}$ is the product of the rings $\mathbb{Z}/q\mathbb{Z}$ where $q$ is a prime power.

Thus, the study of the rep theory of $\text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ is naturally reduced to the study of the rep theories of the groups $\text{SL}_n(\mathbb{Z}/p^k\mathbb{Z})$ for prime $p$. Fixing $p$ and varying $k$, these are grouped together as the (continuous) representation theory of $\text{SL}_n(\mathbb{Z}_p)$, where $\mathbb{Z}_p$ is the ring of $p$-adic integers.

For $n=2$ the irreducible representations of $\text{SL}_n(\mathbb{Z}_p)$ are indeed classified, as appeared in the answere by Jim Humphreys.
To this answer I wish to add the reference to *Uri Onn's paper* in which he classifies all irreducible representations over an arbitrary discrete valuation ring. See also a comment below by A Stasinski.

For $n\geq 3$ the representation theory of $\text{SL}_n(\mathbb{Z}_p)$ is much more complicated and a classification is currently out of reach. However we do know that $\text{SL}_n(\mathbb{Z}_p)$ has only finitely many irreducible representations at each dimension $d$. Denoting this number by $r_d$ we are interested in the study of the sequence $(r_d)_{d=1}^\infty$. On this problem there have been in recent years a remarkable progress on which I wish to report, and this is why I am writing this answer.

For $n=3$ it is shown in by Avni-Klopsch-Onn-Voll
that the sequence $r_d$ grows similarly to $d^{3/2}$. For a precise statement see *their Duke paper*.
Recently Aizenbud-Avni got a uniform statement for all $n$, that is $r_d$ is $O(d^{22})$.
For this see *their Inventiones paper*.
Whether such a uniform bound exists was asked as a question in *Larsen-Lubotzky*.

I would be happy to tell more about the techniques of these remarkable papers, but this answer is already getting too long. You will have to read through the sources.
Let me just mention that in order to study the sequence $(r_d)$ we set the *representation zeta function* $\zeta(s)=\sum r_d\cdot d^{-s}$. A general result of *Jaikin-Zapirain* tells us that this function is rational in the variable $p^{-s}$. The representation growth is then related to the study of the poles of this zeta function. Let me also mention that there is a global theory here, which comes from the representation theory of $\text{SL}_n(\mathbb{Z})$.

**Edit:** As remarked by Alexansder Stasinski (thanks!), for $n=3$ (but not for higher $n$'s) the representation zeta function is explicitly computed in the *Avni-Klopsch-Onn-Voll paper* alluded to above (see Theorem E) and one should be able get out of it the exact numbers of irreducible representation of $\text{SL}_3(\mathbb{Z}/q\mathbb{Z})$ at each dimension, which is closer to answering the OP question. One can also hope to get a full answer using Kirillov orbit method. However, things get messy for higher $n$'s.