After winning a tournament, go to the feast at the same city and talk to one of the ladies. You'll only be able to use that dialogue option once per tournament win, as far as I'm aware.
I'm pretty sure that personality is randomly generated between games, so you'll need to find a local minstrel/skald etc to find out who has what if that's what you're after. Certain personalities are more likely to accept you than others - I had zero difficulty with one who liked to hunt and ride. Other than that, you should probably look for someone whose father you're on good terms with, and who isn't being courted by someone else. Unless you've already got high renown the father won't let you see her if someone better is courting you, but otherwise it's not too hard to get permission.
I must confess that the name plays a role when I'm considering it too, as some names just don't sound right.