Looks like a bit of root fly on them.
I'd say the compost you used was useless.That's probably the biggest problem. Carrots grown in proper compost are a proper colour, nice and clean, not coarse like yours are.
Try a different variety next year. Autumn King generally aren't best in tubs, and aren't harvested until next month usually.
Go for a quicker maturing variety, usually the "nantes" family, and try and protect against root fly.

Carrots grown in a container are likely to suffer periods of drought, followed by periods when they are overwatered, followed by more dryness and so on. This always creates tough, woody roots. Unless you are growing a quick crop of small spring carrots I would always recommend growing in open ground, no matter how little space you have. That way they can root deeply and draw moisture from the ground consistently.

The best carrots are grown in containers using a special compost. When you see such carrots at shows they are grown in drums at least 15" tall as that is the minimum for a good stump rooted carrot.
As Bertie says, watering is the important bit, never letting the compost dry out. This applies to open ground too. If the soil is to dry and then you get a few downpours the carrots will split.
Either way, you must have a stone free sandy soil.