Basically only the initial setup requires you to be online (doh). Make sure you save your account credentials at startup and don't tag "Don't save my account credentials" in the settings.

It's a buggy feature but under normal circumstances, assuming your games are uptodate(not downloading a patch) and you have started them while online at least *once* (this is where it checks the game with Steam), you should normally be able to put it into offline mode, even if you're offline at the time.

There's some caveats and problems for people though. Sometimes something buggers up in Steam and it simply wants to reconnect to internet regardless of what you do. Doesn't mean there is never anything you can do, but it's possible.

A few known cases:
If you shut down your computer with Steam running, there's a chance Windows cuts off the Steam closing down process too quickly and it'll mess something up, which makes Steam need to log in next time. Can be prevented by closing Steam before closing windows.

If you're behind a router and your internet is down it's possible Steam doesn't recognize complete loss of internet and doesn't want to go into offline mode for some reason. This can be solved by turning off your Internet connection in windows, or turning off the router, or pulling your internet cable out of the computer.

Easiest thing though is to simply put it in offline mode anyway, even when you're online, assuming you have no interest in it's online features anyway.

Also, about two computers.
Although it's not possible to log in twice with 1 account, it *is* possible to have one computer with Steam logged in to the account, and the other with Steam in Offline mode and both computers being able to play games of that account at the same time. You won't be able to play anything with multiplayer features connected to Steam though.