This trophy is glitchy. The description says you need 5000 kills with the ray gun but I was a long way past that when the trophy unlocked.
So, you have to get the 5000 in one session. This means you can't turn the Vita off and you can't close the game down. The quickest way to do this is to start at level 1 as you only get grunts meaning there are no wasted kills. I only had this trophy left after completing all others so just used the ray gun to try and kill all the aliens. If you can get to level 8 it's about 200 kills.
This trophy and all the others in this game are made much easier by exploiting a glitch. When you have enough orbs to buy something in the shop, you can buy it once and then keep pressing X to fully upgrade your weapon or defences for no extra cost. This also works for the consumables in the game.