Your character will be asking out those that reply to you, in whatever fashion is appropriate for them! Oh, how adorable! This may not be that positive or healthy of an experience for characters who swing a little darker, of course.
The approaches can be serious, goofy, fluffy, indirect, awkward, unplanned, perfectly executed, whatever you want. Be cute. Be tsundere. Do it in person. In a note. IT'S UP TO YOU.