One doesn't need character theory calculations to decompose the group algebra into irreducibles.
Suppose $G$ is a finite group and $k$ a field in which $|G|$ is invertible. Then Maschke's theorem applies so all finite-dimensional representations are expressible as a direct sum of irreducibles. Write
$k[G]\cong \bigoplus_{V\in\widehat{G}} V^{\oplus m(V)}$
for some unknown multiplicities $m(V)$, as $V$ ranges over irreducibles in $\widehat{G}$. Fix an irreducible $W$, and consider an intertwiner $k[G]\to W$. Such a map is determined by where $1$ is sent, and then conversely $1$ may be sent to any element of $W$. Hence $\dim\hom_{k[G]}(k[G],W)=\dim W$.
On the other hand, using distributivity of $\hom$,
$\hom_{k[G]}\left(\bigoplus_{V\in\widehat{G}} V^{\oplus m(V)},W\right)\cong\bigoplus_{V\in\widehat{G}}\hom_{k[G]}(V,W)^{\oplus m(V)}\cong {\rm End}_{k[G]}(W)^{\oplus m(W)}$
(since $\hom_{k[G]}(V,W)=0$ if $V\not\cong W$) which has dimension $m(W)\dim{\rm End}_{k[G]}(W)$.
Equating $\dim W=m(W)\dim{\rm End}_{k[G]}(W)$ gives us the multiplicities $m(W)$. Hence
$k[G]\cong\bigoplus_{V\in\widehat{G}} V^{\oplus (\dim V)/(\dim{\rm End}_{\large k[G]}(V))}\implies |G|=\sum_{V\in\widehat{G}}\frac{(\dim V)^2}{\dim{\rm End}_{k[G]}(V)}.$
If $k$ is algebraically closed, $\dim V$ divides $|G|$ for each $V\in\widehat{G}$. This fails in general. However there is a weaker version which still holds: $\dim V$ divides $|G|\varphi(\exp G)$ for each $V$. Again, if $k$ is algebraically closed the number of irreducible representations equals the number of conjugacy classes (there is no generic, canonical bijection - instead they are "dual"). This fails if $k$ isn't algebraically closed, or if $|G|$ isn't invertible. And again a weaker version holds: the number of irreducible representations equals the number of $K$-conjugacy classes of $K$-regular elements.
These facts and more should be discussed at the GroupProps Wiki.