How did zeus kill cronus?

Quick answer: Zeus did not kill Cronos. Cronos, like the other Titans, and all his children, is immortal and cannot be killed. There are varying accounts of Cronos' ultimate fate (usually depending on whether a Greek or a Roman told the story). The oldest accounts say that he was imprisoned in the cave of Nyx, away from the other Titans. Eventually he was released and made the king of Elysium (the place where good people go). Other account (particularly Roman) allude to him being released and living in exile in Latium, where he ruled over a Golden Age under the name "Saturnus".

Cronus was told by his father Uranus and Mother Earth that one of his own sons would someday over throw him. Cronus then swallowed the first five children that his wife, Rhea, bore him: Hestia, Demeter, Hera, Hades, and Poseidon. To stop this from happening. He meant to swallow the next child, Zeus, but Rhea tricked him and saved the baby. She took a stone wrapped in cloth as Zeus and gave in to Cronus. Zeus then was raised by nymphs and grew to manhood in secrecy of his father. Eventually, he returned home and became cup-bearer for Cronus. After drinking poison mixed into his drink by Zeus, Cronus threw up his five children and the stone. This is how his siblings came to the world.

After releasing his brothers and sisters, Zeus led a war against Cronus and the other Titans. The war went on for ten years when Zeus, Poseidon & Hades released the Cyclopes from captivity. In gratitude, the Cyclopes gave each of the brothers a weapon. Poseidon received a trident, Zeus a thunderbolt, and Hades the helmet of darkness. They used these gifts to finally defeat Cronus and the rest of the titans. Now that the three brothers were the rulers of all existence they decided to draw lots to determine their domains. Poseidon rule the water, Zeus rule the sky, and Hades rule the underworld.