Battle Grade S is a mixture of two things: the amount of enemies killed, and how long it took you to complete the episode. Most times, the amount has to be greater than 1300-1500 enemies, while the time should be under 10 minutes to be safe. This may not be doable early on, but once you've completed the main Dramatic Log episodes, you should have a comfortable character that you can easily sweep episodes with after it's been upgraded. Return to Free Log and complete episodes, making sure to aim for both of these requirements.
Another option, and this might be better for you: once you've finished the Dramatic Log and have a suitable character, dive into the Treasure Logs and use the Online feature. Pairing up with anywhere from one to three other players may divvy up the enemies slain, but on the Results screen, the cumulative amount will be shown. This will also lower the amount of time you've spent in the episode, almost ensuring you get an S rank unless someone messed up somewhere.