To make things easier, killing a thargoid scout (the smaller alien ships) count towards this achievement (as well as the Heartless achievement). I have not killed a single thargoid interceptor (the big ones).
This can be and was achieved solo in an Alliance Chieftain, no engineering to the weapons or shields, no special anti-thargoid weapons or defences either.
I was consistently able to find the scouts by travelling to a system with a 0 population (in the galaxy map, population count will show up on the left when you highlight a star). I flew around waiting for unidentified signal sources to spawn. After scanning, I would only be interested in Non-human signal source level 4 (the level pops up shortly after the source type). A level 4 generally gave me around 5 to 6 scouts once I dropped out of supervruise but it also meant that there was typically another NPC ship or two, this serve to distract the scouts so that only 1 or 2 will attack you at any one time. A level 5 source seemed to have far more scouts and no distracting NPC’s. A level 6 source would have a Thargoid Interceptor (you can find black boxes and escape pods at these points though)
The scouts are not necessarily easy targets but a couple gimballed multicannons did the job in about a minute of dogfighting. Unguided missiles were very useful if you manage to score a hit (seeker missiles are useless as the Thargoid causes them to miss every time). Watch out for the caustic attack as that will slowly cause your hull strength to degrade (a quick stop at the nearest station will clean your hull though).