How can I upgrade gnome-terminal on Ubuntu 14.04 LTS?

You have two ways to upgrade your terminal from 3.6.2 (stable) to the latest 3.12.3 and read step by step do not run apt-upgrade unless you done installing.

1st solution with Gnome3/staging PPA

sudo add-apt-repository ppa:gnome3-team/gnome3-staging

sudo apt-get update

sudo apt-get install --reinstall gnome-terminal

2nd solution with this PPA:

sudo add-apt-repository ppa:towolf/transparent-gnome-terminal

sudo apt-get update

sudo apt-get install --reinstall gnome-terminal

Though this PPA may require Gnome3/staging PPA to be installed also, this 2nd PPA contains different fixes and patches on RGB schema and transparency, opacity etc.

Notice!

After you add PPAs like Gnome3/staging beware of upgrading as you can even damage your system this is the latest bleeding edge updates and they don't guarantee the stability, so after you install gnome-terminal it's better to purge them e.g.:

sudo ppa-purge ppa:gnome3-team/gnome3-staging

sudo apt-get update

(PPA-purge has to be installed 1stly sudo apt-get install ppa-purge)