Not sure about the other ones, but I can answer this:
The recruitable characters in Part 1 are: Lopez, Tommy, Dante, Jillian, Driver, and Rosie. (All other characters will join whether you want them or not.)
To recruit them:
Lopez: Appears in front of your house on Day 2, between 10 and 12. He pretty much just needs to know you exist, so as long as you go out to help him (or even steal from him, apparently), then he should turn up at the end of Part 1.
Tommy: I think you might need to go out to the crowd outside the Makarov’s house at the end of Day 1, just to let him know you exist, but the important scene is Day 2, between 12 and 2. Help him (and be nice to him), and he should turn up again at the end of Part 1, and may agree to join you, if asked. Note: Empathy is important here. (You can also try to manipulate his want for revenge, which requires less Empathy…)
Dante: Appears in front of your house on Day 2, between 2 and 4. Save him from the zombies. (High Ranged will let you snipe them, but you can also use other methods.) Then he should also turn up again at the end of Part 1.
(Note: all three of the above characters will appear at any point in their time windows as long as your character is at home, so don’t spend the entire time out at the three-story house…)
Jillian: Will turn up at your house at the end of Day 2 with Lyle. It’s best to be kind, but not weak here. Call them out on stuff (Jillian going through your stuff, Lyle faking his blindness), but don’t throw them out. Cook a meal for them. If they like you enough, then they’ll stick around (if not, they’ll abscond with your possessions…), or at least Jillian will. (Lyle will always either leave or die. To save him, since he may be back later, follow them on the morning after, and use Leadership/Intimidation/Persuasion on Dillan when he tries to kill them; fighting may work here, but I’m not sure.)
Driver: Outside the school on Day 3. If you’re a hacker or teen, you can avoid him entirely, but if not, talk your way past him. Don’t try to distract him, as this can lead to his getting zombied. After the school mission, he should turn up again, and can join.
Rosie: When she’s trapped in the auditorium, save her. Like Driver, she’ll turn up after the school mission.
It should be obvious, but don’t kill any of these characters (or Gina, who’ll join in Part 2) if you want them on your team. Also, I know that Jillian may try to kill Rosie and/or Driver after the school mission, but I have no idea how this is triggered, or how to stop it once it starts.
In Part 2, Reverend Church and Gina will turn up. I think all you have to do is not kill them or turn them away. There’s also Sifer, who always dies, but can give you good items when she goes. To meet her, visit the boarded-up house on Day 2 of Part 1.
Also, there are Challenge characters, such as the nephew and Ray, but I’m not sure if they’d count.
@Eiwynn: Would it make sense to add these to the top post? 