Why does Harry Potter say the killing curse to Voldemort in the Battle of Hogwarts?
The very short answer is that Harry never used the killing curse, not even once. During his fight with Voldemort during the Battle of Hogwarts, he casts an expelliarmus spell which conflicts with Voldemort's killing curse. The rebound of the failed spell was sufficient to kill Voldemort.
A red-gold glow burst suddenly across the enchanted sky above them, as an edge of dazzling sun appeared over the sill of the nearest window. The light hit both of their faces at the same time, so that Voldemort’s was suddenly a flaming blur. Harry heard the high voice shriek as he, too, yelled his best hope to the heavens, pointing Draco’s wand:
The bang was like a cannon-blast and the golden flames that erupted between them, at the dead centre of the circle they had been treading, marked the point where the spells collided. Harry saw Voldemort’s green jet meet his own spell, saw the Elder Wand fly high, dark against the sunrise, spinning across the enchanted ceiling like the head of Nagini, spinning through the air towards the master it would not kill, who had come to take full possession of it at last. And Harry, with the unerring skill of the Seeker, caught the wand in his free hand as Voldemort fell backwards, arms splayed, the slit pupils of the scarlet eyes rolling upwards. Tom Riddle hit the floor with a mundane finality, his body feeble and shrunken, the white hands empty, the snake-like face vacant and unknowing. Voldemort was dead, killed by his own rebounding curse, and Harry stood with two wands in his hand, staring down at his enemy’s shell.