This achievement requires quite a bit of skill, but also a lot of luck. The idea is to knock off a couple of shielded planets first, acquire them and then attack only once you have a shield on in order to avoid losing more energy. The part where luck plays a part is for the enemy system to come close to other large systems or stars and crash. Obviously, if such a system is close you can try to attract it towards the enemy and make your life easier. Otherwise, it is nearly impossible to destroy using only your star.
Spawning as a large star is recommended, not neutron, as a neutron star will implode upon losing all energy and not downgrade.
Starting as a neutron star to become a black hole does not work. Neither will a dark star. However, sometimes a dark star will linger around if you spawn in that form and then change to a different type. Conceivably, you could try to draw it towards the enemy system, although I never managed to do that myself.