Hi,
from what I can read from your pic, the file system is damaged, if you have the time, I would suggest you re-install and this time format both the root and home partitions with Ext4!!.
Trying to fix the file system would take a long time and has no gaurantees that it will work!!.
You could try running "dpkg" from the recovery menu if you can get to it!!. That may fix it for you!!. (You must have a cabled network connection, it won't work with Wifi at this level!!)
Regards Wolfman