Once you are already infected, there are few things you can do to detect the virus. Installing or re-enabling the anti-virus will not reveal the virus as it might manipulate the AV software as well.
There are few things that can raise your suspicion though.
First thing is major performance slowdown, especially if you remember that the computer was not as slow and you did not add programs that can slow it down.
Second is what you described. If your personal information is being compromised then it might be a virus on your computer. But remember that it might be other things as well. For example your password for the game was weak or you logged in when connected to a public wireless network or your friend got your password somehow etc.
Third thing is crashes and blue screens. If you experience these a lot, it might be a hardware problem, driver bug, but sometimes it is a virus.