How do you access royal navy archives?

RN records are kept in a number of places, depending on what you are looking for. The best place to start would be the Public Records Office, although the RN Historical Branch in Portsmouth Dockyard keeps many important documents.

The RN website has a list of contacts for all these places

If you are interested in your own personnel records, you would need to raise a Subject

Personnel records for other people, if still alive would not be released. If deceased, records are withheld for 25 years unless release is authorised by the next of kin, although some factual information (name, service number, medals etc) can be obtained on request

Any other information you require can be requested under FoI, although some/all may be withheld on national security or other grounds.