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.