Under no circumstances should you give someone like this information or access to your computer! They will try to sound very convincing but Microsoft never calls you to offer support. (In fact, Microsoft even wants you to know that they don't do this, so they wrote a post on these scams.)
If you receive a call from this person, the best thing to do is hang up immediately. If you have caller ID, make a note of the number they are calling from and report it to your local police department.
If you do give them your credit card information, contact your credit card company and report the incident. If you gave them access to your computer, change your passwords from a safe computer and keep a close eye on your bank accounts if you do any banking online.
Also contact us so we can help you clean up your computer and make sure that there are no more security "holes" where "Alex" can re-access your information.