We study a class of derived representations of a Lie algebroid. The dg-category of these representations enhances the classical category of representations in the sense that the cohomology objects of a derived representation are classical representations. The adjoint complex is canonically an object of our category of representations. Our main contribution is the construction of an extension of the functor of de Rham-Lie cochains to this category, referred to as Crainic-Moerdijk-cochains. We show that, when applied to the adjoint representation, we obtain the Deformation Complex of Crainic and Moerdijk.