Please note that copies of Elite Dangerous purchased from the Epic Game Store MUST be run through the Epic launcher, as detailed in this FAQ. Linking your Frontier account to Steam will NOT add the game to Steam.
When you first start the game from the Epic Games Launcher, the Elite Dangerous launcher will pop up with this message as shown below: 
Click 'Link Account' and you will be brought to a page where you will need to enter your account details: 
Fill out your details as shown on this form and click submit. If you receive the error message that a user with that email already exists, please use the 'Login' button in the Existing Users section.
Once you have done this you will be brought to this page to confirm your email address. 
Once you have reached this page check the inbox of the email address that you submitted on this form, and you should find an email like this: 
On this email please click the link provided to confirm your account creation, once you are done you should be brought to a web page that will confirm your account has been created successfully and it will redirect you to the launcher. if you do not get redirected please open up the launcher again, and you should find that you are now logged in and able to click 'Play'!
If you experience any issues with this process, please do send in a ticket to our Customer Support Team by clicking the button below and the support team will be able to assist you further.