You must get the relationship bar of one of your party members all the way to the end of the blue side. To do this you must do things that the character would approve of. For most characters, you'll generally get friendship points by making choices aligned with the (borrowing from Mass Effect) Paragon path. It is very dependent on each character's background. For example, helping mages will usually earn you points with Anders, but will gain rivalry with Fenris and vice versa. Try to keep a party where all the members have similar views.
By far the easiest party member to get full friendship with is Bethany. Just remember that you can't be a mage if you want her in your party beyond the first stage. From the beginning you'll already have her in the blue side of the bar, and she'll have a quest for you almost immediately after you enter Kirkwall. Generally do the right thing and you'll gain her full friendship in no time.