How did Natsu manage to kill Silver Fullbuster (Gray Fullbuster's Father)?
Natsu didn't kill Silver; in fact, nobody directly killed him. Silver was already technically dead for the entire span of the series starting from the time when Deliora destroyed Gray's home; he was just kept in a "living state" with Keyes' necromancy curse. Silver died (for real) because Juvia killed Keyes, causing the necromancer's power keeping Silver "alive" to disappear, which finally lead to Silver's death.