Character Endings are accomplished by raising a particular character's affinity through Private Actions. You'll have to do one playthrough as Laeticia and one playthrough as Raymond to see them all. There is currently no New Game+.
If you hop around to each town/city during the chapters (and after any major chapter events) and talk to each of your characters, you should get most of them. You can also clean up a bunch (but not all) just before taking on the final boss.
During the non-FMV Private Actions your main will often be asked a question with a multiple choice (timed) answer. Just pick the answer that fits best to the PA character (or use a guide) and you'll add to their affinity as well.
Lastly, you can use Compound IC, level 10, with an Aquaberry and a Roly-Poly Bean to occasionally make 'Tears of Aphrodite', which you can feed to whatever character you want to see the ending for prior to taking down the final boss. I made 10 and fed them all to the one I wanted just to be sure.
Note: As far as I am aware, you can only see ONE character's Character Ending per boss defeat. Simply make your Tears of Aphrodite, save just in front of the final boss area, use the ToA on whomever you want to see, and beat the boss. There are 3 scenes that play every time. Once you've seen these, you can skip and just get to the fourth which is the character ending. After that, either save your cleared game on a DIFFERENT SLOT or press 'B' to skip the game clear save and reload the one you made just before the final boss.