It sounds like the solution you ask for, using your Bitcoin addresses on another computer, isn't really the problem you are having. It sounds like you don't have the complete blockchain downloaded yet. If you moved to another computer, you would have to start downloading from the beginning.
Here is a thread of interest to you: http://bitcointalk.org/index.php?topic=51456
If you do want to move all your bitcoins and account info to a new computer, the file to copy is %APPDATA%\Bitcoin\wallet.dat
. Put it in the same place on the other computer (using the appropriate user name), and run Bitcoin.