First of all, you have to unlock all four chip slots, and then you have to buy 4 chips that boost your health by 5 bars. Equip all four of them to increase the health of your ship by 20 bars.
Secondly, you have to pick up a power-up that increases your health by 20%. Once you pick up the power-up the trophy will unlock. Of course all the power-ups are random, so you might have to grind some levels to get it. I would strongly recommend to do the grinding at the Darkness levels.
Of course the same strategy can be applied to the other three Trophies that require to double the base Attack, Speed and Attack Speed of your spaceship.
Lastly, if you pick up a power-up that increases your health by 15%, it's possible to earn the trophy in a later level if you pick up a second health power-up.