Yellowing Dill Leaves

I am growing some dill in a container on my back deck, and the outer leaves of the dill keep yellowing, eventually dying and drying up. In the meantime, healthy-looking new leaves have sprung up, so the plant seems to be doing all right, but these leaves too eventually yellow and die. The soil is moist but not soggy and it receives sun most of the day. At first I thought it was just transplant shock, but this has been going on now for a month or two. What can I do to help my dill?

If your plant has a lot of leaves, then it would only be normal that the old leaves die, while new ones emerge. If the number of leaves is not getting more there may be a nutrient deficiency, in which case it would benefit from fertilization.
Try fertilizing the dill with a hydroponic fertilizer. When plants do not have enough of an essential nutrient, then to make new leaves they have to take it out of old leaves and this will make the old leaves wither. Since you do not know what nutrient may be missing a hydroponic fertilizer is most useful, because it contains all the nutrients a plant may need.
If this does not help, check the medium around the roots. It should be well draining, but moist. If is thick yellow clay, dig up the plant, keep the roots moist with a wet newspaper while you mix sharp sand and well-rotted manure into the clay, then replant and water in well. Shade the plant with newspapers for a few days and then gradually remove the shading over the period of two days.