Technically you can make any melee character DW now with wardrobe stuff, but if you want to build a legit DW dps character you're looking at either rogue or warrior (more likely rogue).
Plenty of specs for both that allow for DW (dps is calculated primarily by attack power rather than weapon damage) so it depends on what kind of DW spec you want to play. Do you want to be stealthy? Tankier? Do physical damage? Elemental damage? etc.