There are two different versions of the story. Believe whichever one you want to, it still makes for the perfect revenge against an undeserving & corrupt king.
1. A king banished him for being ugly, so Chanakya got his revenge by overthrowing him with Chandragupta.
This is the Buddhist version. When Chanakya was a kid, he had canine teeth. It was a mark of royalty, one day he was meant to rule. His mother was always scared that he would ignore her after getting the throne. For his mother's satisfaction, he broke his teeth.
A grown up Chanakya went to attend an alms-giving ceremony held by King Dhanananda at Pushpapura. Chanakya also had crooked feet. His appearance was not pleasing to Dhanananda's eyes, so he had Chanakya banished.
An angry Chanakya vowed revenge and managed to evade arrest by escaping to the jungle. He somehow managed to befriend Pabbata, Dhanananda's son. While in the forest, he saw a boy play-acting with his friends.
The boy pretended to be king, presiding over a trial of some robbers, where he orders their limbs to be cut off, only to magically heal them later. That boy was Chandragupta. Chanakya took him under his wing.
He saw Pabbata and Chandragupta as options for replacing Dhanananda. So he decided to test them. One day when Chandragupta was sleeping, he asked Pabbata to remove Chandragupta's woolen thread without waking him. Pabbata was unsuccessful. The next night, Chandragupta was given the same task. He succeeded by cutting off Pabbata's head and retrieving the thread and became the chosen one.
It took years to prepare an army for an invasion. They attacked Dhanananda's capital - Pataliputra. But were defeated. They managed to evade capture and one day, under disguise, while wandering the city, they overheard a mother scold her child for eating his cake from the center and wasting the edges.
She compared that to Chandragupta's attack in the sense that he did not capture the neighbouring villages and directly attacked the core. On learning their lesson, they took the right course and captured areas around the capital city first. Soon they marched on to Pataliputra and overthrew Dhanananda.
Chandragupta was appointed king and Chanakya persuaded a fisherman to reveal the location of Dhanananda's treasury and had the fisherman killed. His revenge was complete.
2. A king's servant disrespected Chanakya and threw him out, so he came back with Chandragupta to dethrone him.
According to the Jains, the monks claimed Chanakya was born to be the power behind a throne as he had a full set of teeth. His father, fearing that he might get arrogant broke his teeth. Chanakya, married a Brahmin woman, who was often ridiculed for marrying a poor man.
Chanakya had heard rumours of Dhanananda being generous towards Brahmins. So, seeking donation, he went to visit him. While waiting for the king, he sat on his throne. A servant offered him a different seat. But Chanakya simply placed his water pot on it and did not leave the throne. More offers of other seats were met by more objects being placed on them.