Just write a straightforward letter firstly stating that you are appealing their decision not to award the DCA (enclose a copy of their decision). You are the best advocate for your child and no one knows or can describe their condition and how it affects them and you as a family, better than you. You outline the condition, how it affects the child, their schooling (if appropriate) and how their condition requires more care and attention than another child of a similar age (which is really what qualifies anyone for the payment). Describe the extra that you have to do for him/her. Generally by the time you get a diagnosis they are going to school. Mention if they have a special needs assistant or extra hours from teacher etc. If you spend time repeating work done in school, when at home - how long you spend. Do they have extra tuition outside school. Any aids or equipment you have bought for them. If they have trouble sleeping or require fewer hours than other children and if you are 'on call' or hyper aware because of this. List their medication. Do you work part time to faciliate the child, or not at all, due to the condition? Do you have extra help from friends/family? How does it impact on any other children you might have? Behavioural problems - Lay it all out, in bullet points if necessary. Then finish by saying that you are evidencing the above in the form of diagnosis or report from whatever the appropriate medical professional is that made the diagnosis. Would also be useful if you had a letter of support from school and GP. The more evidence you have of what you are saying the better. Keep copies of everything.
It is basically their medical opinion versus 'yours'. Generally speaking your medical professionals are better qualified to determine the nature of your child's condition, than the SW people as you have psychologist/psychiatrist etc - think at initial stage with SW, its just a desk assessment so emphasis should be placed on your professionals' eminent qualifications and experience to make such a determination. Also google Social Welfare Appeals and have a look at other appeals of a similar nature - should get a better idea of whats required and the successes of other claimants.
I would think a lot of applications have to be appealed in the current climate. There seem to be a lot of unsuccessful applications since DCA moved from HSE to SW. Do your best. Am sure it will be good enough. Good luck.