I've got an installation of Jenkins on a Ubuntu 14.04.5 LTS which I needed to update. In order to do so I did the following:

  1. Download the new version (2.107 at the moment of writing this post). In /home/user/backup/ execute:

    wget http://updates.jenkins-ci.org/download/war/2.107/jenkins.war

  2. Backup the old version (Jenkins war is installed in /usr/share/jenkins):

    sudo cp /usr/share/jenkins/jenkins.war /home/user/backup/jenkins-2.8.war

  3. Overwrite the old jenkins.war:

    sudo cp /home/user/backup/jenkins.war /usr/share/jenkins/jenkins.war

  4. Restart jenkins

    sudo service jenkins restart

Now, you're all set and you can start updating the plugins.

HTH,