It seems natural to work with an algebraically closed field of characteristic $p$, or, less restrictively, a splitting field of characteristic $p$ for $G$. For example, any field containing the primitive $m$-th roots of unity, where $|G| = p^{a}m$ and $p$ dos not divide $m$, so I assume now that $k$ is algebraically closed of characteristic $p.$ We have entered the realm of modular representation theory, whose theory was first extensively developed by Richard Brauer. By now the basic theory is covered in numerous texts (eg by Alperin, by Curtis-Reiner).

The indecomposable direct summands of the group algebra $kG$ are the so-called projective indecomposable modules (sometimes called principal indecomposables). The number of isomorphism types of these is the number of conjugacy classes of elements of order prime to $p$ of $G.$ Each of these has a simple socle (the socle of module is its largset semisimple submodule) and a unique maximal submodule (so a simple head, or largest semisimple quotient module). Because the group algebra is a symmetric algebra, the socle and head of $P$ are isomorphic. If $P$ is one of these projective indecomposable modules, and has (simple) socle $S$, then up to isomorphism, $P$ occurs ${\rm dim}_{k}(S)$ times as a summand of the group algebra $k[G].$ Hence $S$ occurs $dim_{k}(S)$ times (up to isomophism) as a summand of the socle of the regular module $kG.$ Also, $S$ occurs ${\rm dim}{k}(P)$ times as a composition factor of the regular module $k[G].$ Every simple module for the group algebra $k[G]$ has a projective cover, and these all occur as direct summands of the group algebra. As Mariano remarks in his comment, the theory of non-projective indecomposable modules is much less transparent. The results over non-splitting field can be recovered using Galois theory and Clifford theory. However, modular representation theory is much richer and diverse than this brief description allows for. These basic facts (and much much more) were all known to Brauer, perhaps sometimes in different language, and these are just the beginnings. If $|G|$ is not divisible by $p,$ the projective indecomposable $P$ and its socle $S$ are the same module, and the theory degenerates to the semisimple case, which is much like the characterisic zero situation.

indecomposable representations, then not only it is not obvious that all of them are in the group algebra: it is in fact generally false (for there are infinitely many of them) On the other hand, if by irrep you meansimple representations, the situation is more hopeful; for example, in the extreme case where $G$ is a $p$-group, there is only one simple representation, the trivial one. But now you have to decide what exactly you want "live in F[G]" to mean, as the algebra is no longer semisimple. – Mariano Suárez-Alvarez♦ Jul 18 '12 at 19:51