From what I understand it requires a Fusion Core, but where can I get one? Is there a certain type of UFO of special mission it comes from? Is any specific research required?
Building the Blaster Launcher requires that you:
The Battleship can be difficult to shoot down, so send your best interceptor after it if you can, and be ready to use your boosts. The Battleship salvage run is also a large map - plan for an engagement that's just slightly shorter than assaulting the alien base. There's no mistaking this for one of the smaller craft, it's probably about double the size of the supply ship. Try to shoot it down with an EMP cannon, to maximize your chances of getting 2 Fusion Cores intact.
PROTIP: you can force a Battleship to appear by ignoring a UFO that is on a "scout target" mission - after a few days a Battleship will appear over the same country on a "destroy satellite" mission. This way you can farm Fusion Cores for Blaster Launchers.
Alternatively, if you have the Slingshot DLC, that series of missions culminates in disabling a battleship well before you'd normally encounter one, with two guaranteed intact fusion cores as well as a lot of Elerium and alloys.
You can get them from a Battleship (will show up as "Very Large" if detected), though I can't say if you can miss them in a recovery mission due to damage/mean RNG or not.