Add all the ingredients except the chicken in a small bowl. Whisk to combine.
Place the chicken in a large glass bowl or gallon-size storage bag and pour in the marinade. Toss everything together to coat, cover, and refrigerate for 2 to 6 hours. (Do not marinate longer than 8 hours.)
To grill: Preheat the grill to medium-high heat and lightly oil the grates with vegetable oil. Grill the chicken for 5 to 10 minutes per side depending on the thickness of the meat until cooked through and the internal temperature reaches 160-165°F on an instant read thermometer.
To cook on the stovetop: Heat 2 tablespoons of oil in a large skillet over medium-high heat. Add the chicken and cook for 5 to 10 minutes per side, depending on the thickness of the meat until cooked through and the internal temperature reaches 160-165°F on an instant read thermometer.
To bake: Preheat the oven to 425 degrees F. Arrange the chicken in a single layer on a large foil-lined baking dish. Bake the chicken until the internal temperature reaches 160-165 degrees F on an instant-read thermometer. Chicken thighs and tenders need about 15 minutes. Medium to large breasts need 20 to 25 minutes.
To make in the slow cooker: Add all ingredients into a large slow cooker plus ½ cup chicken broth or water. Mix together to coat the chicken, cover, and slow cook on low for 6-7 hours or on high for about 4-5 hours.
Transfer the chicken to a plate, loosely tent it in foil, and let it rest for 5 minutes before serving.
Notes
Don't have achiote paste? You can leave it out completely or substitute regular paprika in its place. It won't quite have the same flavor, but it will still be delicious!