If you find an entry in the GRO indexes then you can send for the certificate from the GRO, although in the case of those early army birth regsitrations you won't usually get very much information. The various Army births come from a range of different sources, and some of them are baptism registers.
No matter what the original source is, the certificate you get will be in standard certificate format, and not a copy of the original document. Sometimes there is quite a lot more on the original, but unfortunately there is no way you can get to see it. If you find two or more children in a family and they all have exactly the same reference, it's not worth buying a certificate for more than one of them.
There may be records at The National Archives relating to the family, but they are quite unconnected with the GRO records. You already have the man's service record, which is the single most useful document, but sometimes details of wives and children appear in the regimental records, such as muster rolls.
Hope this helps
Mean_genie