Preheat an outdoor grill or grill pan on medium high.
Mix together the hot sauce, brown sugar, paprika and cayenne in a small bowl. Transfer 1 tablespoon of the mixture to a large bowl and whisk in the mayonnaise; set aside the rest of the sauce. Add the chives, celery and onions to the bowl with the mayonnaise and toss well.
Sprinkle the salmon with salt and pepper. Brush the grill grate lightly with oil. Lay the salmon on the grill, skin-side up, and cook until distinct grill marks appear and the salmon releases easily from the grate, 2 to 3 minutes. Turn and brush the fish with some of the reserved sauce. Continue to cook the fish, brushing the pieces periodically with the sauce, until the salmon fillets are glazed and just cooked through, 13 to 15 minutes more.
Transfer the fillets to individual plates and serve with the celery slaw.