Computer 1 does not have to be on-line for the iTunes library to be shared. Computer 1 and 2 are connected to each other by virtue of the fact that they are both connected to the Router. With the right account setup on both machines and file sharing turned on, ALL the files on BOTH the computers can be accessed.
The absolute easiest, down and dirty, way to do that is as follows:
Turn on File Sharing in the System prefs pane on C1 and C2
On C1, go to the GO menu and select Network. You should see the name of C2 in the resulting window. Double click and you should be asked for a username and password to connect to C2. Enter the name and password of the User of C2 (Not your name and password, the name and password of the user of C2) If all is well, you should now see an icon on the desktop with the name of C2 on it. Open that and you have access to everything.
Repeat the process from C2 but use the name of the user of C1.
What you have actually done is to log onto the other machine, as the user/admin of that machine and NOT as yourself.
This is NOT the recommended way to do it but it works quickly. The correct way is to create an account on the other machine and log into that account. The problem with that is that you can only access files that are in your account and none of the files that are in the other users account. There are ways around it but it gets v e r y involved.

hope this helps