When Gawain first arrives at Arthur's court, it's with two of his brothers and his mother, Morgause, who has come to spy on Arthur. It's not exactly a hero's entrance. Gawain's father, King Lot of Orkeney, is at war with Arthur over the throne, which doesn't bode well for Gawain's relationship with his king.
On the other hand, Gawain is Arthur's nephew through his mother. Being related to the king is a definite advantage for Gawain, and despite his father's belligerence, it's not long before Gawain is a Knight of the Round Table, fighting at Arthur's side at jousts.
Unfortunately, though, Gawain just can't seem to do anything right. In his first adventure, he fails to show mercy to a knight who asks for it, then kills the knight's lady by accident. Big mistake. As punishment, he must ride to Arthur's castle wearing the lady's head around his neck, which is totally humiliating, not to mention totally gross. In his next big adventure, the lady he's supposed to be protecting rides away from him in disgust when he refuses to lend aid to a captured knight. Then Gawain promises another knight to help him win his lady, and ends up sleeping with her instead. Another big no-no.
So, to sum up, Gawain is not courteous to ladies, and he doesn't keep his word. Which means that, as a Knight of the Round Table who has sworn to do these very things, he doesn't exactly excel. We might blame some of Gawain's actions on his personality. After all, the narrator tells us that Gawain was a "passyng hote knyght of nature," which, in medieval medical theory, designates him as choleric, or passionate. He's given to intense rages or fits of temper (590.41). We'll summarize: he's impulsive, hot headed, and a bit of a jerk.
But we can't just write it off as a personality problem. Gawain's biggest problem actually stems from his family. Even after King Lot's war with Arthur ends, his sons, including Gawain, continue the family's aggressive behavior toward the family of King Pellynore, who killed King Lot. Gawain, along with his brothers, ambushes Pellynore's son, Lamorak. This kerfluffle ends with Lamorak being stabbed in the back, which is definitely not how a knight is supposed to go.
Even just participating in an event with such a dishonorable ending smears Gawain's good name. And it's not like Gawain has to participate, as the example of his brother, Gareth, proves. No one but Gawain is to blame for his bad behavior and his poor handling of family drama. And it has to be said that he doesn't learn from his mistakes. He shows similar hot-headedness in his pursuit of Launcelot, who inadvertently kills Gareth in a battle.
At least at the end of his life poor Gawain tries to make good. He repents of his feud with Launcelot and writes a letter to reconcile with him. This final act is in keeping with Gawain's pattern, which is to do something bad, then feel bad about it and apologize. After all, it's not that Gawain's a bad guy; it's just that he can't seem to get the hang of this whole chivalry thing.
He's stuck in a world of familial honor and personal vendettas that doesn't match up with the chivalric world of Arthur's court, where knights must put honor, courtesy, and loyalty to the king above even family. That's something that Gawain just can't seem to do. In the end, it's this quality that brings about not only his own downfall, but that of Arthur's whole court. Poor guy. If only he'd taken some anger management classes.