The first weapon you should be able to fully upgrade is the Service Pistol, and it only has three upgrades that you need to purchase. I was able to upgrade every slot on the Service Pistol at level 19 with only a few challenges completed, mostly in the first challenge category. Just purchase the following three upgrades for the pistol:
Armat A7 Sound Suppressor
Armat Laser Targeting System
A27 High Velocity Rounds
Note: You may also need to change the weapon skin to something other than the default skin.
EDIT: After doing some checking, it seems that all of the pistol upgrades are related to leveling up and/or completing missions. Completing challenges is not necessary for this one. Also, it seems that the weapon skins do not affect this achievement, and you can unlock the achievement regardless of weapon skin.
If you do this with the Service Pistol, you will also unlock this achievement: