Easiest way is to hook it up as an extra drive to your tower pc (if you have access to one). http://www.datarecoverytutorial.com/recover-data-from-failed-laptop-notebook/ scroll down to middle of page, it's a toshiba sata drive in the example. Or you could connect it with a cable to a usb port (see link below for pic)
If you only have access to another laptop then you will need a sata to usb cable like this one: pic of cable as it is a 2.5 hard drive you do not need an extra power source as the usb will provide the power.
You will be able to browse the contents of the old harddrive and copy the files you want to the other harddrive. If you got a caddy you could continue using the old drive as an external Harddrive.