Add all ingredients except the chicken in a small bowl. Whisk to combine.
Place the chicken in a large ziploc bag or baking dish and add the marinade. Close the bag, massage together to coat, and refrigerate for 2 to 6 hours. (Do not marinate longer than 8 hours, or the chicken will get mealy and begin to break down.)
Preheat the grill to high heat for direct cooking. Grease the preheated grill grates to prevent sticking by wiping them down with a folded paper towel that's been soaked in vegetable oil.
Remove the chicken from the marinade and shake off any excess liquid. Grill the chicken skin side down over direct heat for about 5-7 minutes, with the lid closed as much as possible, until the skin is crispy and golden. Flip and grill for an additional 5-7 minutes, until the meat is firm and the internal temperature registers 165°F.
Serve pollo asado with warm corn tortillas or flour tortillas and Mexican rice.
Notes
Achiote paste: You can find it in most Mexican grocery stores or even in the international aisle of some big-box grocery stores. No achiote? Smoked paprika with a pinch of turmeric gets you close on the color.
Oven method: Preheat the oven to 400°F and line a baking sheet with aluminum foil. Arrange the marinated chicken on top in a single layer and bake for 20-25 minutes, flipping halfway through, until the internal temperature reaches 165°F. Broil the chicken for 2-3 minutes at the end to give it some extra char and color.
Stovetop method: Cook in a grill pan or skillet with a little oil over medium-high heat, 5 to 7 minutes per side, until 165°F.
Storage: Store leftovers in an airtight container in the fridge for up to 4 days, or freeze for up to 3 months. Reheat in a skillet over medium-low with a splash of water or broth, or microwave in short bursts so it doesn't dry out.