Fury is unlocked during Chapter 2, when you find your first Purity Seal. To gain Fury, simply damage enemies. When the Ultramarines icon in the bottom left corner of the screen is filled with gold, click both thumbsticks to enter Fury. While in Fury, you can simply mash the button, as your damage output is increased, and you will heal with every hit you make.
A few good place to work on this are near the end of Chapter 5, where you must defend an elevator from waves of Orks and in the second part of Chapter 2, after you find your Purity Seal that unlocks Fury Mode.
For the Chapter 2 method, simply go to Chapter Select under Single Player, select Act 1, Chapter 2, and the section labeled "Trenches". Play through the Chapter until you reach the Digga cutscene where Ork machines bust up through the ground into the Imperial Guard trenches, then continue around the next few corners until you see a mounted Heavy Bolter. This is the location you want, and you can rack up quite a few kills here. Once you've use your Fury meter up and finished off the Orks, reload your last checkpoint, run back around the corner, and rinse and repeat.