A nutrionist would say yes, drinking Diet Coke is bad for you. It has no nutritional value whatsoever. They would probably agree however, that is is better than regular Coke since it has no calorie content. There is a lot of caffeine in Coke so it would be better if you could keep it at one per day.
I like what kilikina has said here, but I do want to point out that beyond having no nutritional content, sodas (especially the diet kind) are filled with artificial flavors and sweeteners. You're better off avoiding the whole mess altogether. Try mixing seltzer water with juice, if you like something fizzy.
Artificial sugars are bad for you because they trick your body into thinking you are drinking sugar and when you do not get that sugar you "crash" and your body makes you crave sugary foods to replace the sugar it thought it already got, which can lead to weight gain because you eat something high in sugar to make up for the perceived sugar intake.
The carbonation in sodas contains chemicals that break down calcium in your body, which can lead to bone softening and bone loss if you consume enough.
Caffeine can lead to addictive behavior and in come cases nervous behavior disorders.