(For simplicity I will discuss $G = SL_n$. Making the necessary modifications to $GL_n$ shouldn't be hard.)

The most important result here is Steinberg's restriction theorem, which tells you that the restriction of $L(\lambda)$ to $G(\mathbb{F_q})$ is simple if $\lambda$ is of the form $\lambda = \sum \lambda_i \omega_i$ with $0 \le \lambda_i \le q-1$ for $1 \le i \le n$ ($\omega_i$ are the fundamental weights). (Let us call this set of dominant weights $X_q$).

The second most important result here is Steinberg's tensor product theorem, which tells you that for any highest weight $\lambda$ if we write

$ \lambda = \lambda_0 + p \lambda_1 + \dots + p^a \lambda_a$

with each $\lambda_i \in X_p$ then $L(\lambda) = L(\lambda_0) \otimes L(\lambda_1)^{Fr} \otimes \dots \otimes L(\lambda_a)^{Fr^a}$ (where $M^{Fr}$ denotes the Frobenius twist of a module $M$, i.e. inflation with respect to the Frobenius $Fr : G \to G$.) This allows you to reduce to the case $q = p$.

Hence one needs to understand the modules $L(\lambda)$ for $\lambda$ in the box $X_p$ decompose. As you mention, these are the simple socles of induced modules $\nabla(\lambda)$.

For $n = 2$ this is easy: each $L(\lambda) = \nabla(\lambda)$ for $0 \le \lambda \le p-1$. Hence $L(\lambda)$ is realised as homogenous polynomials in $k[X,Y]$ of degree $\lambda$ (with natural $SL_2$ action).

For $n = 3$ things get more complicated. (Here the relevant $\nabla(\lambda)$ can have either 1 or 2 composition factors, and one gets the answer using the Jantzen sum formula. (See Jantzen's "Representations of algebraic groups").

For $n = 4$ things get even more complicated! Here I think one can still get the answer from Jantzen's sum formula, but this seems more of an accident here.

In general there is a character formula (due to Lusztig) which holds for large $p$ (see Jantzen's book). In the case of $n = 4$ this is known to hold for any $p > n$.

Representations of Algebraic Groups, which Geordie refers to. (See my notes people.math.umass.edu/~jeh/pub/weyl.pdf) This makes no difference for the composition factors. The case $n=2$ goes back to classical work of Brauer-Nesbitt, while $n=3$ was done (modulo Steinberg's 1963 paper) by a student of Curtis in his 1966 thesis, and $n=4$ is due to Jantzen in 1974 for $p>n$ (the Coxeter number). For "small" $p$ lots of uncertainties remain, but computers help a bit. $\endgroup$Finite Simple Groups II(Academic Press, 1980). For $GL_n$ you just have to multiply by powers of det, which requires mainly some bookkeeping. $\endgroup$Modular Representations of Finite Groups of Lie Type, which appeared in 2006 as London Math. Soc. Lecture Note Ser. 326 (Cambridge U. Press). Almost nothing there is original, but it includes some treatment of special cases including general linear groups. You might find the 460 references useful. (See also the revisions on my homepage.) $\endgroup$