Typically, the first instance of an acronym is written out in full, followed by the acronym in parentheses. If you are generating the body of text yourself, use this format:
... private contractors with top secret clearance had access to Government Communications Headquarters (GCHQ) databases.
Future references to this acronym can be simply made as GCHQ now that you've defined what that means.
If you are quoting another work, you should include your text after the acronym in brackets (as you have done in your question, but with an added space):
... private contractors with top secret clearance had access to GCHQ [Government Communications Headquarters] databases.
An alternative, of course, is to include the full term in a glossary at the end of your publication (if it's long enough to warrant a glossary) and just use the acronym by itself in your text:
... private contractors with top secret clearance had access to GCHQ databases.
Side note: I'm not 100% certain, but I think that Top Secret is the proper name of a clearance level and should be capitalized.