To prepare the chicken, combine the chicken, olive oil, lemon juice, vinegar, Greek seasoning, and salt in a sealable container. Marinate the chicken in the refrigerator for at least 30 minutes or up to overnight.
For the tzatziki, mix together the yogurt, grated cucumber, garlic, lemon zest, lemon juice, dill, salt, and black pepper in a medium bowl. Refrigerate the tzatziki until ready to serve.
Heat a 10-inch (25 cm) or larger cast-iron skillet over medium-high heat. Add the marinated chicken to the skillet. Cook the chicken for 3 to 4 minutes on each side, or until browned and its internal temperature reaches 165°F (74°C).
To make the red wine vinegar dressing, whisk together the olive oil, vinegar, oregano, and salt in a small bowl.
To assemble the bowls, divide the cooked chicken among four individual serving bowls. Top with the diced cucumber, halved tomatoes, sliced onion, Kalamata olives, and crumbled feta cheese. Drizzle the red wine vinegar dressing over the bowls and add a dollop of tzatziki just before serving.