Daucus carota Full Sun USDA Zones 4 -10
Carrots grow well in containers, they do best in cooler temperatures as opposed to the balmy hot summer weather. They are a hardy crop that should be planted in early spring.
Compost added to the containers is a good idea for outdoor plants. Some people use worm castings, if you have access to these - go for it. See Worm Composting.
Carrots require larger amounts of water than many other popular edible garden vegetables, they are not tolerant of drought or prolonged hot weather - especially in their later stages.
In hot summer weather pots heat up significantly quicker than ground soil, so keeping them well watered and mulched to retain moisture is a must.
Mulch on the soils surface, such as wood chips or decorative bark is not only aesthetically appealing inside or outside it is beneficial to the health of your plants.
Trays under the pots are used for house plants to catch excessive water, they aren't necessary with tomato or other outdoor plants. The trays will hold too much moisture at the base of the roots, for too long and could easily lead to root rot.
Make Sure Your Plants are located where they can get adequate light. Carrots will tolerate partial shade but thrive in full sun, 6 - 8 hours daily. During hot weather it is advisable to move them to a shadier location to avoid the excessive heat they do not fare well in.
Smaller carrots such as Oxheart, Romeo or radish style carrots are best for growing in pots, they are only 2 - 3 inches long and generally have a quick maturation period. Standard varieties will work also, with the correct container size - yes - size does matter. Some standard length carrots will be slightly stunted when grown in containers.
You may have to thin seedlings after they sprout in order to maintain a healthy separation for the plants to develop. Generally about 3 inches apart is optimal, this varies slightly with some varieties. You can generally fit in 3-4 crops of carrots in a growing season after which you should refurbish some of your soil with a new mix, I generally remove the top half of the soil and replace it, then every 3rd or 4th time I'll replace the entire mix.