List with all visible meta data in AFTER UPDATE trigger
You could start with ORIGINAL_LOGIN(), SUSER_SNAME(), HOST_NAME() and 'APP_NAME()'. 

However as @JeroenB said you may have better luck creating a trace or extended events session.  You can get the above information and more that way.  Make sure to restrict down the trace/EE session to just the one table and you shouldn't see a lot of overhead.  You could certainly stick with the trigger though and see if that gets you the information you need.  The main thing you could get out of the other options (that I would want to see) is the actual command being used.  I haven't found a way to get that out using a trigger I'm afraid.