It is important to remember that conger eels have strong jaws and sharp teeth, so they will make short work of a normal hooklength. That means you need a strong hooklength and Malcolm suggests 150lb mono.
The actual rig is a simple running leger with the 150lb hooklength carrying a size 8/0 or 10/0 O’Shaughnessy pattern hook. The lead weight is clipped on a simple zip slider, which is threaded either directly on the reel’s mono mainline or on a mono leader tied to braid mainline. Make sure you place a bead either side of the swivel. The mainline and hooklength attach to a strong rolling swivel, so choose a quality brand with a breaking strain of at least 100lb.
You'll need size 8/0 or 10/0 hooks, strong swivels and zip sliders