This tutorial is of three trophies, they are:

(1+4)x
(5+5)x
(10+1.5+3.5)x

The reason is, you can achieve all three at the same time.

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Trophy Description(s) -

(1+4)x - Get a multiplier of 5x
(5+5)x - Get a multiplier of 10x
(10+1.5+3.5)x - Get a multiplier of 15x

Trophy Type(s) -

(1+4)x - Bronze
(5+5)x - Silver
(10+1.5+3.5)x - Gold

You gain multiplier by destroying enemies. The faster you kill them the quicker the the multiplier will increase. Dying will reset the multiplier as will waiting too long between attacking an enemy, or picking up/saving a human.
Each difficulty is capped at the following multipliers:

Rookie - 5x
Experienced -10x
Veteran - 15x
Master - 20x

The higher the difficulty you play on, the faster you will accumulate the multiplier meter. However, it will also reset significantly faster as well. This makes gaining a large meter of even 8 or 9x+ on Master difficulty rather challenging. The best way to go about unlocking this trophy is by playing with Nemesis on Veteran difficulty, on the Decima level. Its ability to use boost almost indefinitely allows you to get out of trouble and preserve your life, while also instantly killing tougher enemies. This coupled with homing missiles and medium kill speed is the perfect combination for high multipliers.

Play through the entire level with one goal in mind: preserving your multiplier meter. If you die half way through just restart the level. The end of phase 2 and all of of phase 3 are the most dangerous sections, so try and save all of your bombs for then.
Dont immediately kill every enemy because you risk running out of things to shoot and thus resetting your meter. Try and always leave a few guys alive if you are between wave spawns, and pick up/save humans to refresh it as needed. The goal is to get to the boss with as high a meter as possible while saving up an Overdrive.
During the boss fight destroy every single last grey cube it creates, down to the smallest one. After it it creates its third clone immediately use your overdrive on it. Doing this will also unlock the trophy O to the D if you haven't already obtained it. Be sure not to shoot the grey cube before using overdrive because this will mess up your damage. You'll know the cube is about to replicate when it flashes the red square around itself. When done correctly it will massively boost your already high meter. Its very possible to get to 15x by doing this level from start to finish.