In BBC Merlin, Agravaine is the brother of Arthur's mother, Queen Ygraine, and he resents Uther for his role in causing Ygraine's death. It's understandable that Agravaine would want Uther to be overthrown. What makes less sense is that, rather than wanting to install Ygraine's son in Uther's place, Agravaine prefers to put Morgana there and is even willing to kill Arthur to achieve his goal. Why would he take Morgana's side, at great risk to himself, when he is neither a fellow magician nor her blood relative?
