How to get all logged in users to SQL Server database for this week?
You have a few options, some of which were already pointed out.

First, though, if these events haven't been captured so far there is no "historic" way of obtaining the data. There may be ways to get some of the data but chances are you won't get all or even most of it.

Going forward, however, it seems you only want those who successfully connect to the instance. This can be complicated if something such as connection pooling with a sql login, but I'll leave that as something for you to decide on how you want to track it back to the users.


Server Side Trace


Even though traces are deprecated they still work well. If it's just a successful login needed then this can be fairly low overhead. It has well known functions to load into tables and do your analysis.

http://msdn.microsoft.com/en-us/library/cc293613.aspx


Extended Events


Since you only need the successful logins, there is an event "sqlserver.login" that can work well for you. Send it to an event_file target and it's fairly simple as well. There are also functions to load the files and do your analysis.

http://technet.microsoft.com/en-us/library/bb630317(v=sql.105).aspx


Logon Triggers


Probably my least favorite option but here for completeness. The logins could be logged to a table, but why re-invent the wheel? Trace and extended events will do this for you.

http://msdn.microsoft.com/en-us/library/bb326598.aspx


SQL Audit


I didn't bunch this with extended events, even though in reality it leverages the same framework. This is an enterprise only feature, and while it can do this and is targeted for items such as this... I feel most aren't running enterprise edition.

http://msdn.microsoft.com/en-us/library/cc280386.aspx