Combine the lemon juice with 1/2 cup olive oil in a resealable plastic bag; seal and shake. Add the tuna steaks and store in refrigerator to marinate while the barbecue preheats.
Preheat barbecue for high heat and lightly oil the grate.
Whisk the brown mustard, mustard powder, sugar, vinegar and 1/3 cup olive oil together in a bowl; stir in the dill.
Remove the tuna from the marinade and discard the marinade; cook the steaks on the preheated barbecue until the fish flakes easily with a fork, about 6 minutes per side. Drizzle the steaks with the mustard and dill sauce to serve.