A term that may fit in the scope of this problem is "generic character table", character tables of a whole family of groups of Lie type.
Example: Generic character table of $SL_2(q)$, $q = 2^f$
| CharactersRepresentations | $I$ | $U$ | $S(a)$ | $T(b)$ |
|---|---|---|---|---|
| Trivial | $1$ | $1$ | $1$ | $1$ |
| Steinberg | $q$ | $0$ | $1$ | $-1$ |
| Principal indexed by $k$ $k=1 \dots q/2 − 1$ | $q+1$ | $1$ | $\epsilon^{ak}$$+$$\epsilon^{−ak}$ | $0$ |
| Discrete indexed by $l$$l=1 \dots q/2$ | $q-1$ | $-1$ | $0$ | $−\eta^{bl}$$−$$\eta^{−bl}$ |
where $ \epsilon = \exp (2πi/(q − 1))$, $η = exp (2πi/(q + 1))$.
The generic character tables of the groups of Lie type $D_5(q)$ and $E_6(q)$ are still unknown, let alone $E_7(q)$ and $E_8(q)$.
And even if we worked out the whole character table, there is still a gap from the character table to representations. To see this, you can try to derive the representations from the character table above.