Representation theory of the general linear group over a finite prime field

One classical source for the case $n=2$, with somewhat old-fashioned notation for some of the related groups, is a paper by Richard Brauer and his student Cecil Nesbitt: On the modular characters of groups, Ann. of Math. (2) 42, (1941). 556–590. In this very special case, the actual modules are not too difficult to describe in terms of symmetric powers of the natural 2-dimensional module. But for arbitrary $n$ there are still mostly unknowns.

For a modern survey which includes most of the relevant references (especially to work of Steinberg, Jantzen, Lusztig), see my LMS Lecture Notes 326: Modular Representations of Finite Groups of Lie Type (Cambridge, 2006), in particular Chapter 19. As in the study of the ambient algebraic groups, a parametrization of modular irreducibles by highest weights is readily given in terms of highest weights, which can be translated if desired into the language of partitions for general and special linear groups. See Jantzen's book Representations of Algebraic Groups (2nd ed., AMS, 2003). (Steinberg's twisted tensor product theorem from 1963 then shows how to deduce results for finite fields larger than the prime field.) While these particular groups are often much easier to study than arbitrary groups of Lie type, we still seem to be far from a complete understanding. And the partition viewpoint may not be too helpful.

But the details about dimensions and possible constructions of the modules are still largely unknown, apart from $n=2,3,4$. As Jantzen's book demonstrates, the dimension problems are well-organized and for large primes such data can in principle be computed from Lusztig's viewpoint in terms of Hecke algebras for certain affine Weyl groups. But as Geoff remarks, construction of the actual modules is quite problematic, while even the computations of degrees and such is usually beyond reach in practice.

Concerning tensor products and other constructions, little has been worked out except for very small $n$.

I would suggest James-Kerber: Representation Theory of Symmetric Groups (Encyclopedia of Mathematics and its Applications, vol.16) Addison-Wesley, 1981 The last chapter deals with the modular representation of general linear groups, and in particular, Exercise 8.4 gives the construction of all irreducible $\mathbb{F}_p[GL_n(\mathbb{F}_p)]$-modules.