So my opponent often attaches this condition to one of my characters. I will pick that character to die first, but where does the attachment go? Does it go in my discard pile or my opponents?

Don

Your opponent's , since he is the owner and the controler of the attachment

Kelemvor said:

This is the important bit. Cards will always go the their owner's out-of-play areas (dead pile, discard pile, shadows, hand, deck). The owner of a card is the player who had it in his deck at the start of the game.