Langlands Dual Groups

The Chevalley construction mentioned by Steven Sam from Lusztig's notes can also be found in Springer's textbook "Linear algebraic groups" (on pages 164-165 of the second edition). It's beautiful and clear and elementary (no perverse sheaves!) and direct (no Lie algebras, which wouldn't work in finite characteristic, and no reduction to simply connected groups). If you want to construct representations of the dual group directly, then geometric Satake is a good idea. For the classical Langlands conjectures, what one cares about are not representations of the dual group but elements of it (or a little more precisely, maps of Galois groups into it). For such purposes, the Chevalley construction by generators and relations is just what Dr. Langlands ordered.

You can construct the dual group in a combinatorial manner as follows: Reductive groups are classified by their root datum. There is an obvious duality on the set of all root data, and the dual group is the reductive group with the dual root datum.

You can see Wikipedia for the notion of the root datum.

One way to "construct" the Langlands dual group is via the geometric Satake isomorphism, which gives an equivalence between the category of representations of the Langlands dual group and the category of G[[t]]-equivariant perverse sheaves on the affine Grassmannian. From the category of representations, the group can be reconstructed via Tannakian duality, i.e., as automorphisms of the fibre functor. I suppose whether or not this is an explicit construction depends on your point of view.