Without spoiling too much...
Demise: Pretty straightforward. He loves conflict and pretty much just wanders around to see what he can find and fight. He saw the power Hylia had over him and likely assumed the Triforce was the source of her abilities. He probably wants to obtain the Triforce to best her and then continue on his rampage.
Dethl: Dethl pretty much just wants to feed and preserve itself. Dethl was present at the creation of the Triforce, and without saying to much believes that obtaining it will essentially end its hunger.
Vaati: Vaati is likely entertained by the notion of being able to do whatever he wants, whenever he wants. Judging by the stories of the Triforce, he probably wants to use it to expand his abilities like the Goddesses. Vaati suffers from an inferiority complex, what better way to compensate.
Majora: Majora obviously has a bruise with the Goddesses, I suppose I can leave it at that.
I don't really want to say too much about the Triforce either. We do know it can be destroyed as that is what Majora does near the end of the game, so it's safe to say it's not all it's cracked up to be.