He is not a typical rat, and his personality is more like that of a human, as he walks on two feet instead of four because he doesn't like the idea of his paws touching both the ground and the food he eats. What he thinks of his talent is a bit of modesty, as shown when Linguini asked him if he could actually cook. Towards the fact that he is not really a typical rat, this concerns his father excessively, and always ends up in an argument with him to do with his ways.
His overall dream is to become a chef. He is incredibly skilled and talented in cooking (possibly beyond the likes of Gusteau), never failing to impress diners (including feared top food critic Anton Ego) with his excellent quality dishes, to the point that Linguini (who is controlled by Remy) becomes the star chef of the restaurant.
At the end of Ratatouille, Remy's dream comes true in which he becomes the head chef at the La Ratatouille bistro in Paris.