How does Gawain's dislike of Launcelot to lead to Arthur's downfall?

Answer 1

Lancelot was having an affair with Arthur's wife Guenevere. When they were discovered Guenevere was sentenced to death by burning on the stake. Lancelot then saves her,kills one of Gawains brothers in progress and escapes to his castle Joyous Garde.

While Arthur was willing to let both of them go unpunished he is pressured by Gawain, angry over his brothers demise and thirsting for revenge, to besiege Lancelot's castle. When Lancelot is finally defeated Arthur exiles him to France.

However, during Arthur and Gawain's absence Mordred rebells and crowns himself King. When Arthur returns, Mordred has grown too powerful and the war against him results in the destruction of Camelot.

Short : Because of Gawains wanted revenge on Lancelot for killing his brother, Arthur was forced to leave Camelot, giving Mordred the chance to claim the throne, and banished Lancelot to france. Without Lancelot at his side Arthur was unable to defeat Mordred without the destruction of Camelot

Answer 2

Lancelot was carrying on an adulterous affair with Guinevere.