Adrenaline is earned through killing the creatures of the games world.
After you die for the first time you'll
. From.here it is a matter of following the mission pathway without taking any damage whilst killing everything in sight.
LT half way down will use the gun normally and after you find the uograde holding will let you use a charged up shot for a one hit kill on the lower leveled enemies. This seems to build it quicker but using circle to dash out of the projectiles way is very handy. If you get them to aim straight at you whilst on the ground you are able to jump over them using the X button.