Sign into Xanga and visit a blogring where you're currently the Leader.  In the upper left module, it should say:

Members 
» Invite Members
» Approve Members
» Manage Members
» Your Membership

Click on "Manage Members", and you should see a list of all the members in your blogring.  Under each member name is a link labeled "Remove". 

Find the member you'd like to delete, and click on the "Remove" link underneath that username.  This will cause a confirmation window to popup.  If you click "OK", you'll be asked to signin.  Once you've confirmed your identity, the member will be deleted from the blogring.

If you'd like to make sure the member can't just rejoin the blogring, you may want to turn off auto-approval (if it is already on).  Please see:

How do I turn off auto-approval of new members, so that new members have to apply to join?
http://help.xanga.com/howturnoffautoapproval.htm