When you talk to a character in School Mode, there are two options: spend time together, and use a Trip Ticket. Spending time together increases their friendship level, which goes towards unlocking pages of their report card. Using a Trip Ticket increases their heart level (if you answer correctly), which goes towards viewing their character ending.
The friendship level is shared with Story mode, so you can also increase it by replaying chapters. Give them presents they like and answer the multiple choice questions correctly to increase friendship level and max report cards as fast as possible. There's a good table of who likes which presents here.
http://ouendanl.tumblr.com/post/35460955577/parbles-dangan-ronpa-present-table-x-is-bad
Now before you can view any character ending in School mode, you have to max out ALL the report cards. So even if you maxed Sayaka's report card, for example, you won't be able to view her ending until you max ALL the report cards.
When you use a Trip Ticket, you pick a place to go and then get a multiple choice question. You may also get a Trigger Happy Heart event, it happens randomly. I *think* you need to view this event to view the character ending, but I'm not sure. I haven't found an English guide to which answers are best, but you can figure it out from Japanese wikis and Google translate. You can talk to Monokuma to see everyone's heart level.
http://danganronpa.wiki.fc2.com/wiki/お出かけ
On the 51st day, you will be able to talk to characters at max heart level and view their ending.