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. This theorem also gives a construction, as does B&K's "Schur-Weyl duality for higher levels".