For a single group of relatively small order, computer methods are available, as pointed out by John Wiltshire-Gordon. For a general theoretical approach, there is much less to go on in terms of methods. But in my old expository article in *Amer. Math. Monthly* 82 (1975), I did reference the work done (before the Deligne-Lusztig era) on representations of finite special linear groups over finite fields. Besides the character theory worked out by Frobenius (and the American H. Jordan independently), there are only a couple of serious attempts to describe the actual representations. The two references I could find are:

S. Tanaka, Construction and classification of irreducible representations of the special linear group of the second order over a finite field, Osaka J. Math., 4 (1967) 65-84

S. I. Gel'fand, Representations of the full linear group over a finite field, Math. USSR-Sb., 12 (1970) 13-39

Neither of these seems to have given much insight into more complicated finite groups of Lie type, but taken on their own they are worth looking at. I'm not sure what is currently available online, but the second article appears in a fairly standard translation journal. Beyond the simple groups of rank 1, it seems quite challenging to say anything concrete about the matrix description of arbitrary irreducible representations. Though there is the underlying Harish-Chandra philosophy for Lie groups which certainly replicates here at least in the Deligne-Lusztig character theory.

Some added comments:

1) The paper by Tanaka is freely available through the Euclid Project
here. This is closest to the question asked about $\mathrm{SL}_2(\mathbb{F}_p)$, but like Gelfand's work is fairly uniform over any finite field. To pass to the projective group is usually straightforward, by taking into account which representations are trivial at the central element $-I$.

2) At the 1971 Budapest summer school, Sergei Gelfand gave an exposition (in English) of his work, published in the proceedings some years later: Representations of the general linear group over a finite field. *Lie groups and their representations* (Proc. Summer School on Group Representations of the Bolyai Janos Math. Soc., Budapest, 1971), pp. 119–132. Halsted, New York, 1975.

3) While it's easy to pass from a special linear group to its simple quotient, it's much trickier to pass from a general linear to a special linear subgroup (even on the level of characters), so Gelfand's approach may not be so directly useful here. The concrete work in rank 1 also seems to be of little help in higher ranks but has been exploited in work on representations over local fields.

4) In the rank 1 case, roughly half of the irreducible representations (or characters) are easy to construct using induction from a Borel subgroup; even for finite groups, a functional description of induction is traditional here. But the big problem is to pin down the cuspidal or discrete series representations (called "analytic" by Gelfand). Both Tanaka and Gelfand use essentially analytic approaches to the problem, via notions such as "Bessel function" over a finite field.

5) To get real representations, other people have suggested approaches. The well-known character table provides some guidance here.