The answer depends a lot on what kind of description of the radical you ask for. This family of groups of Lie type has been well-studied from the viewpoint of modular representation theory in the defining characteristic (with reference also to the ambient algebraic groups). Even the somewhat degenerate case $p=2$ fits well enough into the general pattern for odd primes.

It's easy to work out explicitly the $p$ irreducible modular representations, for instance using the characteristic 0 model of spaces of homogeneous polynomials in two variables having degree $<p$. Work of Brauer and others filled in the structure of their projective covers in the group algebra; these have very few composition factors. So in this special case you can write down as explicitly as you want all the dimensions involved, including the dimension of the radical (and eventually even its Loewy series). Here are some of the fairly straightforward references, though the story gets far more complicated for groups of higher rank and even for larger finite fields than the prime field:

J.E. Humphreys, Representations of $SL(2, p)$. Amer. Math. Monthly 82 (1975), 21–39.

J.E. Humphreys, Projective modules for $SL(2, q)$. J. Algebra 25 (1973), 513–518.

Henning Haahr Andersen; Jens Jørgensen; Peter Landrock, The projective indecomposable modules of $SL(2, p^n)$. Proc. London Math. Soc. (3) 46 (1983), no. 1, 38–52.

The answer depends a lot on what kind of description of the radical you ask for. This family of groups of Lie type has been well-studied from the viewpoint of modular representation theory in the defining characteristic (with reference also to the ambient algebraic groups). Even the somewhat degenerate case $p=2$ fits well enough into the general pattern for odd primes.

It's easy to work out explicitly the $p$ irreducible modular representations, for instance using the characteristic 0 model of spaces of homogeneous polynomials in two variables having degree $<p$. Work of Brauer and others filled in the structure of their projective covers in the group algebra; these have very few composition factors. So in this special case you can write down as explicitly as you want all the dimensions involved, including the dimension of the radical (and eventually even its Loewy series). Here are some of the fairly straightforward references, though the story gets far more complicated for groups of higher rank and even for larger finite fields than the prime field:

J.E. Humphreys, Representations of $SL(2, p)$. Amer. Math. Monthly 82 (1975), 21–39.

J.E. Humphreys, Projective modules for $SL(2, q)$. J. Algebra 25 (1973), 513–518.

Henning Haahr Andersen; Jens Jørgensen; Peter Landrock, The projective indecomposable modules of $SL(2, p^n)$. Proc. London Math. Soc. (3) 46 (1983), no. 1, 38–52.

ADDED: The general structure of a group algebra (or other finite dimensional algebra) is studied in the traditional way using idempotents in the classical 1962 book by Curtis and Reiner Representations of Finite Groups and Associative Algebras. The idempotents generate left ideal summands (principal indecomposable modules) and survive in the semisimple quotient when the radical is factored out; sometimes these can be described explicitly, as in the case of symmetric groups. In your example of a family of groups of Lie type, split over the prime field, the structure of the group algebra over that field extends naturally to an algebraic closure where comparison with algebraic groups is possible.

In the case of a group algebra, the key information tends to involve the representation theory of the group over the given field. It may or may not be helpful to look for generators of the radical (as a two-sided ideal), but I don't know of any substantial results for this family of groups. The dimensions and module structure are transparent, however.