For $S_n,$ one can construct all the irreducible representations through the young diagrams. Is there any natural construction for the irreducible representation of $G\wr S_n$ (G is a finite group)?

It depends what you mean by natural. In Theorem 4.3.34 of James and Kerber's book "The Representation Theory of the Symmetric Group" a complete list of irreducible representations is obtained via Clifford theory. However this is probably not what you're looking for.
– Jay TaylorJul 29 '13 at 6:50

@Munees: Besides the older symmetric group literature, the more recent work on rational Cherednik algebras might be suggestive. But as Jay points out, it depends on what is meant by "natural".
– Jim HumphreysJul 29 '13 at 14:01