Heat 1 teaspoon olive oil in a skillet over medium heat. Add onion; cook and stir until translucent, about 5 minutes. Stir in diced tomatoes, green tomatoes, capers, basil, salt, and pepper. Cook, stirring occasionally, until flavors combine, about 5 minutes.

Season tilapia fillets with salt and pepper.

Heat 1 tablespoon olive oil in a large skillet over medium heat. Add tilapia fillets; cook until flesh flakes easily with a fork, about 3 minutes per side. Transfer to a serving platter.

Pour lemon juice into the skillet; swirl to pick up browned bits off the bottom of the skillet, 30 seconds to 1 minute. Pour over fillets.

Spoon tomato mixture over fillets before serving.