You will gain access to this near the beginning of the second chapter; Double Trouble. After you destroy the Hermann and Heinrich Capital Warships you will be told to return to the ISS. Don't worry about leaving your Xerg, it will retain all upgrades and you can retrieve it after you're finished with the Spitfire.
Dock at the ISS and change ship - the achievement will unlock straightaway. The Spitfire has superior lasers and a regenerating Heavy cannon, although it is slow and can be overwhelmed by Wotans it is great for accumulating some Brunnhilde and Alberich kills.