First you need to determine the ip address of the computer that goes online you are interested in. This can be done in different ways, much depends on the specific situation. For example, if you sent the letter, examine the title - it always has information about the ip address of the sender. Under the title here refers not to the subject, and comes along with a text service information. For example, a mail service Rambler to view the header, open the email, click "more actions" - "message Headers".
A properly configured computer never "climbs" to the network. If you see that the connection icon in the system tray and then becomes active, check the network connection by typing in the command prompt netstat –aon and press Enter. To open a command line (console) navigate: "start" - "All programs" - "Accessories" - "Command prompt". In the column "Foreign address" you will see all the connections of your computer.
Suspicious ip address should be checked for one of the network services provider of the service who-is. For example, here: http://www.nic.ru/whois/. You should know that at best you will be able to determine the ISP through which the owner of the ip you are interested in coming on line. If this address against you were carried out any illegal action you can report it to the provider, specifying the exact time, your ip address, and describing the situation. In any case, any data about your customer will not tell, this information is sent only to law enforcement.
If you could determine the ip address you are interested in the person, the only real way to gather about him some data associated with the hacking of its computer, which in itself is illegal. Such actions can be justified (from the point of view of morality, but not law) only if this person lied to you or committed other illegal acts, in this case legal means you nothing was not achieved. In this case, you have the right to defend themselves.
Locate the network program Metasploit, read the information available on its application. There are versions of the program both under Windows and under Linux, it's a legitimate application for system testing. Using Metasploit, you can check remote computer for the presence of vulnerabilities. If they are, you will have the ability to penetrate the attacker's computer and gather the information you need.