Since you only care about the completely reducible case, I'll assume $K=\mathbb{C}$.

The easiest way that I know of to construct irreducible $S_n$ representations is a special case of the construction  [here][1]. In particular, symmetric group is a quotient of the (degenerate) affine Hecke algebra appearing there (obtained by setting $x_1=0$). If you take $\mu=0$ in the link, then the skew shape $\lambda/\mu$ is just $\lambda$ and the module that is constructed is the irreducible representation $S_\lambda$.

This recovers the character formula $s_\lambda=\sum_T x^T$, where the sum is over all standard tableaux of shape $\lambda$ (as described in Macdonald's text, for example).


  [1]: https://mathoverflow.net/questions/22515/practical-ways-to-get-skew-schur-functions