You may know that Skype does not have any option to look on the old chat history online and the only way is through the installed Skype application. So you may require backing up the chat history when you are planning for OS re-install or you want the chat in another computer.
Type %APPDATA%\Skype in RUN to open Skype’s Appdata folder, there you will able to see the folders in the name of usernames used to login in that local system. Make sure you have exit the Skype application from tray or you can choose to sign out. Then just copy the folder with your username to your desired location for backup.
1. RUN > %APPDATA%\Skype
2. find the folder which is same as your username.
3. copy and paste it in your backup location.
You can use the same folder in another computer to have the chat history there in another computer.
If you are using any cloud file sync service (like Dropbox, SkyDrive, Google Drive, etc.) between different computers, you may use that to sync that between different computers and backup as well.
In case you are concerned about security about using Skype in shared computer, you can delete your Skype username folder to erase all trace you have had in that computer related to Skype.