Well, to understand how this problem is wild it may be useful to contrast it with the situation of finite reductive groups where we do have a classification statement. The first part of this post considers the reductive case and then after the cut I give some information about $\mathrm{U}_n(q)$. Hopefully some of this is clarifying or useful for your question.

Assume $\mathbf{G}$ is a connected reductive algebraic group over an algebraic closure $\mathbb{K} = \overline{\mathbb{F}_p}$ of the finite field of prime order $p>0$. Furthermore, let us assume that $F : \mathbf{G} \to \mathbf{G}$ is a (generalised) Frobenius endomorphism of $\mathbf{G}$ admitting an $\mathbb{F}_q$-rational structure $G = \mathbf{G}^F$, which is a finite reductive group.

Now, we want to "sensibly" parameterise the irreducible characters of $G$. In general this means that the parameterisation should be given, as far as possible, in terms of data from the algebraic group $\mathbf{G}$ and should not depend on $q$. However, a priori it is not clear at all that such a classification can be achieved. It is one of the great feats of finite group representation theory that Lusztig managed to obtain such an elegant classification theory. We roughly recap this below.

Assume $\mathbf{T} \leqslant \mathbf{G}$ is an $F$-stable maximal torus of $\mathbf{G}$. To every irreducible character $\theta \in \mathrm{Irr}(T)$ ($T = \mathbf{T}^F$) Deligne and Lusztig have defined, using $\ell$-adic cohomology, a virtual character $R_{\mathbf{T}}^{\mathbf{G}}(\theta)$ of the group $G$. We call this a Deligne--Lusztig character of $G$. The following curcial fact is known about these virtual characters:

- The character of the regular representation of $G$ is a sum of Deligne--Lusztig characters. In particular, every $\chi \in \mathrm{Irr}(G)$ occurs in some $R_{\mathbf{T}}^{\mathbf{G}}(\theta)$.

This is somewhat surprising as this is not true of every class function on $G$ (unless $\mathbf{G} = \mathrm{GL}_n(\mathbb{K})$)! With this one has a chance to at least bunch the irreducible characters together but to do this we need more information about the Deligne--Lusztig characters.

Let $\mathbf{G}^{\star}$ be a dual group of $\mathbf{G}$ and let $F^{\star} : \mathbf{G}^{\star} \to \mathbf{G}^{\star}$ be a (generalised) Frobenius endomorphism such that $G^{\star} = {\mathbf{G}^{\star}}^{F^{\star}}$ is a dual group of $G$. We will denote by $\nabla(\mathbf{G},F)$ the set of all pairs $(\mathbf{T},\theta)$ where $\mathbf{T} \leqslant \mathbf{G}$ is an $F$-stable maximal torus and $\theta \in \mathrm{Irr}(\mathbf{T}^F)$. The group $G$ acts naturally on this set by conjugation and we denote the orbits of this action by $\nabla(\mathbf{G},F)/G$. Conversely we denote by $\nabla^{\star}(\mathbf{G},F)$ the set of all pairs $(\mathbf{T}^{\star},s)$ where $\mathbf{T}^{\star} \leqslant \mathbf{G}^{\star}$ is an $F^{\star}$-stable maximal torus and $s \in T^{\star} = {\mathbf{T}^{\star}}^{F^{\star}}$ is a semisimple element. The group $G^{\star}$ also acts naturally on $\nabla^{\star}(\mathbf{G},F)$ by conjugation and we denote by $\nabla^{\star}(\mathbf{G},F)/G^{\star}$ the orbits of this action. With this notation we have the following result:

- We have a natural bijection $\nabla(\mathbf{G},F)/G \to \nabla^{\star}(\mathbf{G},F)/G^{\star}$ which satisfies $(\mathbf{T},1_T) \mapsto (\mathbf{T}^{\star},1)$.

If $(\mathbf{T},\theta)$, $(\mathbf{T}',\theta') \in \nabla(\mathbf{G},F)$ are in the same $G$-orbit then we have $R_{\mathbf{T}}^{\mathbf{G}}(\theta) = R_{\mathbf{T}'}^{\mathbf{G}}(\theta')$. In particular, if $(\mathbf{T},\theta)$ corresponds to $(\mathbf{T}^{\star},s)$ under the above bijection then we may simply write $R_{\mathbf{T}^{\star}}^{\mathbf{G}}(s)$ for $R_{\mathbf{T}}^{\mathbf{G}}(\theta)$. With this in hand we may state one of the most important theorems concerning Deligne--Lusztig characters.

- Assume $(\mathbf{T}^{\star},s)$ and $({\mathbf{T}^{\star}}',s')$ are not in the same $G^{\star}$-orbit then $R_{\mathbf{T}^{\star}}^{\mathbf{G}}(s)$ and $R_{{\mathbf{T}^{\star}}'}^{\mathbf{G}}(s')$ have no irreducible constituent in common.

Now, for any semisimple element $s \in G^{\star}$ we denote by $\mathcal{E}(G,s)$ the set of all irreducible characters $\chi \in \mathrm{Irr}(G)$ such that $\chi$ is a constituent of $R_{\mathbf{T}^{\star}}^{\mathbf{G}}(s)$ for some $F^{\star}$-stable maximal torus $\mathbf{T}^{\star}$ containing $s$. The set $\mathcal{E}(G,s)$ is called a Lusztig series of $G$ and by the above we have a disjoint union

$$\mathrm{Irr}(G) = \bigsqcup_{(s)} \mathcal{E}(G,s)$$

where the union runs over all $G^{\star}$-conjugacy classes of semisimple elements of $G^{\star}$.

Let us now assume that the centre $Z(\mathbf{G})$ of $\mathbf{G}$ is connected (similar statements hold when $Z(\mathbf{G})$ is not connected but this is more complicated to state). One of the most amazing parts of Lusztig's classification result is that, under this assumption, there exists a bijection

$$\mathcal{E}(G,s) \to \mathcal{E}(C_{G^{\star}}(s),1)$$

where $C_{G^{\star}}(s)$ is the centraliser of $s$ in $G^{\star}$. Note that, as $Z(\mathbf{G})$ is connected we have $C_{\mathbf{G}^{\star}}(s)$ is a connected reductive algebraic group. As $F^{\star}(s) = s$ we have $F^{\star}$ induces a (generalised) Frobenius endomorphism of $C_{\mathbf{G}^{\star}}(s)$ and so $C_{G^{\star}}(s) = C_{\mathbf{G}^{\star}}(s)^{F^{\star}}$ is a finite reductive group.

The crucial part of Lusztig's classification result is given by the following result:

- Assume $\mathbf{H}$ is a connected reductive algebraic group and $F : \mathbf{H} \to \mathbf{H}$ is a (generalised) Frobenius endomorphism of $\mathbf{H}$. Let $\mathbf{T}_0 \leqslant\mathbf{B}_0\leqslant \mathbf{H}$ be an $F$-stable maximal torus and Borel subgroup of $\mathbf{H}$. This data determines a Coxeter system $(\mathbf{W},\mathbb{S})$ where $\mathbf{W} = N_{\mathbf{H}}(\mathbf{T}_0)/\mathbf{T}_0$ and $\mathbb{S}$ is a set of Coxeter generators determined by $\mathbf{B}_0$. By our choices $F$ induces an automorphism $\gamma : \mathbf{W} \to \mathbf{W}$ which stabilises $\mathbb{S}$ (i.e. it is an automorphism of the Coxeter system $(\mathbf{W},\mathbb{S})$). Lusztig has then shown that there is a bijection $$\mathcal{E}(H,1) \to X(\mathbf{W},\gamma)$$ where $X(\mathbf{W},\gamma)$ is a set whose definition depends only on $\mathbf{W}$ and $\gamma$.

This is somehow the truly amazing thing. That these irreducible characters $\mathcal{E}(H,1)$ can be parameterised "independently of $q$". This gives us the classification that we desire. In fact, it is really only the classification of the semisimple conjugacy classes of $G^{\star}$ that really depends on $q$. Note that this mimics the classification of the conjugacy classes of $G$. In general a conjugacy class of $G$ is a product $(s)\mathcal{O}$ where $(s)$ is a semisimple conjugacy class of $G$ and $\mathcal{O}$ is a unipotent conjugacy class of $C_G^{\circ}(s) = C_{\mathbf{G}}^{\circ}(s)^F$. The unipotent conjugacy conjugacy classes are parameterised "independently of $q$" and depend only upon the action of $F$ on the root system of $C_{\mathbf{G}}(s)$.

Let us now consider the case where $\mathbf{U}_n$ is the group of unitriangular matrices in $\mathrm{GL}_n(\mathbb{K})$ and $F : \mathbf{U}_n \to \mathbf{U}_n$ is the Frobenius endomorphism $F(x_{ij}) = (x_{ij}^q)$ so that $\mathbf{U}_n^F = \mathrm{U}_n(q)$. One may now take the strategy for finite reductive groups and apply it to our situation here. In particular we would like to do the following:

- Break $\mathrm{Irr}(\mathrm{U}_n(q))$ into series $\mathcal{F}_i$ such that the irreducible characters in each $\mathcal{F}_i$ can be parameterised "independently of $q$".

The theory of supercharacters gives a way to determine such series $\mathcal{F}_i$ (see "Supercharacters and Superclasses for Algebra Groups" by Diaconis and Isaacs). In particular, these are given by the transitive closure of the condition that two irreducible characters occur as constituents of a supercharacter. These series have a nice combinatorial description but in contrast to the case of finite reductive groups once cannot parameterise $\mathcal{F}_i$ independently of $q$. In this sense the problem is wild. See this paper by Marberg for a recap on the supercharacter theory of $\mathrm{U}_n(q)$

http://arxiv.org/pdf/1005.4150v4.pdf

Another approach to the character theory of $\mathrm{U}_n(q)$ is given by the theory of character sheaves on $\mathbf{U}$. This theory (conjectured by Lusztig) was developed by Boyarchenko and Drinfeld. In Theorem 1.13 of

http://arxiv.org/pdf/1006.2476v3.pdf

Boyarchenko gives a classification of the irreducible characters of $\mathrm{U}_n(q)$ in terms of minimal idempotents of the $\mathbf{U}$-equivariant bounded derived category of $\overline{\mathbb{Q}_{\ell}}$-constructible sheaves on $\mathbf{U}$. This may also give a way to see why the classification of such characters is wild by showing that the classification of such minimal idempotents is also a wild problem.

One also has a way to break the irreducible characters up into series based on the theory of $L$-packets. This approach is similar to the Lusztig series encountered in the case of finite reductive groups.