Classifying all irreps of an algebra like a W-algebra is a fool's errand, so I'm assuming that's not what you mean.

For finite dimensional irreps, there are 2; for $\mathfrak{sl}_n$, the number is always the number of standard tableaux of the corresponding Young diagram.  This is Theorem C in Brundan and Kleshchev's [Representations of shifted Yangians and finite W-algebras](http://front.math.ucdavis.edu/0508.5003).  This theorem also gives a construction, as does B&K's ["Schur-Weyl duality for higher levels"](http://front.math.ucdavis.edu/0605.5217).