Ok, so all you need to do for this trophy is to simply keep kicking ass until you level up.
:: TIPS::
To make this a little easier (quicker) to do, instead of using 1 weapon all the time you should change it up as each weapon will need attachments unlocking, and by unlocking attachments for weapons you gain exp points.
So change up your game a little by switching weapons, secondaries etc and you'll start to level up quicker.
This same process can also be used for prestiging and becoming a 'Master Prestige' (if you choos to do so)
Until then... Gun up, and kick ass ;)