There are two ways to go about this. One is to scan the entire drive with data recovery software such as Easeus, Minitools, or Photorec. The other is to correct the corrupted file system.
Depending on how bad the corruption is, a chkdsk can make things worse and therefore more difficult to recover the data. I believe Easeus will let you recover up to 2 GB for free. If you have a spare external drive, you can create a recovery folder on it using Parted Magic. In the folder you can click and select run terminal here as root then type photorec, all lower case, at the terminal prompt. When you run PhotoRec from the recovery folder any files found would be recovered to it but the downside of PhotoRec is everything recovered is in real time and all files recovered have generic names. The files would need to be opened and renamed accordingly.
Another option is to use the ntfsfix option in terminal and then see if the drive will mount. Per your image the command would be
If the command says it was successful you should be able to mount sda5 per this guide. See second image.
If ntfsfix does not work then you could run chkdsk from a Hirens CD but as I previously posted, depending on how bad the file system is corrupted you could be making it more difficult to recover any data.
Edit: In order to run Easeus or Mintiools Data Recovery you would need to pull the drive, attach it to another computer, and run the software on the recovery computer.
Edited by JohnC_21, 11 May 2016 - 01:37 PM.