Combine lemon juice, olive oil, Dijon mustard, garlic, red bell pepper, salt, and pepper together in a bowl. Set aside 1/4 cup of the marinade to use for basting. Place chicken breasts in the bowl and stir to coat. Marinate at least 20 minutes in the refrigerator.

Preheat an outdoor grill for high heat and lightly oil the grate.

Remove chicken from marinade and shake off excess; discard the remaining marinade in the bowl. Cook chicken on the preheated grill until no longer pink in the center, about 6 to 8 minutes per side, basting occasionally with the reserved marinade. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).