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