When a PC is reduced to 0HP they fall Unconscious and they start making Death Saving Throws at the beginning of their turns. Once they fail three death saves, they die (PHB pg.197).
Attack rolls against an Unconscious character have advantage and any attack that hits the character is a Critical Hit if the attacker is within 5 feet of the character (Unconscious condition, PHB pg.292).
If an Unconscious character takes damage while at 0 HP, they automatically fail one death saving throw, or 2 death saves if the damage is from a critical hit. Massive Damage can still outright kill the character so damage should still be rolled and if it equals or exceeds their max HP then they die (PHB pg.197).
Also, since unconscious creatures are usually prone, the advantage gained from the PC being unconscious is cancelled out by disadvantage if an attack is made from further than 5 feet away.
So the first hit drops the PC to 0 Hit Points and they fall to the ground Unconscious. Then if the second attack hits, and the attacker gets a critical hit, the PC automatically fails 2 death saving throws - or automatically fails one death saving throw if it isn't a critical hit.