For the medal, although they do not hold records you could try sending a photograph of both sides with a request for help to either the Greenwich Hospital see http://www.grenhosp.org.uk/
or the Royal Hospital School see http://www.royalhospitalschool.org/
or the National Maritime Museum that has a gallery dedicated to the original school see http://www.rmg.co.uk/visit/exhibitions/on-display/the-royal-...
The National Archives has something that could relate to him in ADM 73/172/85.
ADM 73 = Admiralty: Royal Greenwich Hospital: Pensioners and School Admission Papers, Out Pensions Pay Books and Miscellaneous Registers: School Admission papers.
ADM 73/172 = [surnames] Bin – Blai. (described at item level)
ADM73/172/85 = John Blackmore. When admitted to Greenwich Hospital School: Not Stated.
Parents' names: Benjamin Brandon alias Blackmore.
Applicants surname is stated as Brandon alias Blackmore.
Pupils at the school often enlisted in the Royal Navy and gained rapid advancement, you could try searching the NAs documents online collections for a rating's or officer's records see http://www.nationalarchives.gov.uk/records/our-online-record...
There are likely to more records at Kew from the school but I think you will have to make a personal visit or hire a researcher.