I just did that this last May. On my birthday too!
It's pretty fun, but if it's your first time at
Disneyland it will be really weird.
Basically, you can attend one of the pre-parties with your school if they choose to do one - California Adventure Blast Off, ESPN Zone, etc. They will keep you busy until Disneyland opens at 10 pm. Your school will then be escorted to the check in area where you wait in REALLY cramped conditions for the security check.
Then you go in the park and do whatever you want!!! I absolutely recommend shopping early in the night, because in the morning it is impossible to find anything to buy or to check out (lines fill up the entire stores). They do sell some special Grad Nite only stuff, but not a whole lot.
You can ride rides, dance in clubs, watch fireworks, etc. They have all modern music and almost everything is open. Finding Nemo Submarines and Autopia are not running, among a couple other things. But all the Mountains are open and all the Fantasyland rides are up too. We even saw people on the carousel!
I don't think there are characters out, but you can also enter to win stuff from Honda. They have a car, a motorcycle, and a scholarship. Might as well put in your name if you have time.
You won't even be tired the whole time because they have the park all lit up and music playing. But afterwards, yeah... you will be sleepy! It ends at like 6 am but you can start leaving at like 5 am.
It does get cold in the morning, but just take a sweatshirt and that's all you will need. The night doesn't seem cold up until like 4 am.