Being able to fully control the growing medium is one distinct advantage of growing vegetables in containers. The best compost for carrots in containers is a rich potting soil that is intended for vegetables, such as the Coast of Maine Vegetable Planting Soil available at Walmart. Do not think about using garden soil in pots as it could harbor diseases and also tends to hold too much water. Mix in a handful of homemade compost into the mix to further boost the nutrients.
Rather than adding some general-purpose fertilizer, which may provide too much nitrogen, it is best to add more phosphorus to the soil as that nutrient helps strong root growth. One fantastic way of how to add phosphorus to soil is using bone meal, such as this Natural Organic Bone Meal available at Burpee.