How do I ensure my save file from Chapter 1 carries over to Chapter 2?

Completion data is generated when you see the credits of Chapter 1. (Go to sleep in your bed at the end of the game.)

I'm on a different computer from the one I played Chapter 1 with. Can I play Chapter 2?

You can play Chapter 2 without save data from Chapter 1. As long as you generally remember what happened story-wise, you'll be fine.

So is Chapter 2 the rest of the game? This is it!?

There are planned to be more chapters than Chapter 2. However, we've only completed Chapter 2 so far, so please be patient.