How did the painting become enchanted or whatever?

He simply wishes it would age instead of himself. I think he might have "exchanged his soul" for it but I can't recall, someone probably knows for sure.

I think it was a "Be Careful What You Wish For"/"To The Power of Suggestion" thing.

he made a comment about selling his soul to remain young looking forever when they first laid eyes on the painting

When they're in the studio in the film, Dorian thinks for a moment, and admits he'd give his very soul to be young and beautiful forever. When he does that, you hear a quiet sort of inhale sound, like something taking it's first breath... which is meant to be the painting coming to life and being the visual representation of Dorian's soul.

In the novel it's far more dialogue-heavy, and it takes about two pages for anyone to say anything, so I don't remember exactly when Dorian chooses to give up his soul, but he does say that he's jealous of the painting for staying perfectly beautiful while he slowly ages and withers. So in the book it's more of a trade, in the movie it's, as they say, 'nailing their soul to the devil's altar'