To upgrade a weapon, you need to fill the outline of it in the lower right hand part of your screen three times. It gets harder with each increase, but it is easy enough to do.
To make this achievement easier, I suggest you upgrade five 'easy' weapons. For this I suggest the Kar 98k, the MP40, the Thompson, and the M1 Garand. The Colt M1911 .45 pistol also counts so you could upgrade all the weapons before unlocking any other weapons. I suggest you keep blitzing through the first mission. It's the shortest mission in the game, and the easiest. However, doing the second mission isn't a bad idea at the same time. Play through the game using these weapons most and you'll get the achievement pretty easily.
I suggest you use the pistol for most of the first mission as it features the weakest enemies and therefore makes it easier to use the pistol effectively. Below is a list of the five weapons, and what way I recommend completing them:
Colt M1911 .45
The MP40 arguably gets the best final upgrade, so if you get it later in the game you'll get more use out of it due to the tougher enemies. It's a dagger melee attack, should you be wondering.