To upgrade a weapon, you must first reach the required amount of money (the purple numbers in the top left of the screen) and then hold to open the gun menu and us to select the gun you want to upgrade, then to upgrade it.
Pistol 1 - $500 (Fire rate + laser sight)
Pistol 2 - $1200 (3-round burst)
Shotgun 1 -$1000 (Range + Flashlight)
Shotgun 2 - $3000 (Increased Spread + Damage)
Assault Rifle 1 - $1500 (Shoots faster, seems more accurate too)
Assault Rifle 2 - $4000 (Dual wield!!!!)
Minigun 1 - $1800 ( I have no idea, but it's a minigun!)
Minigun 2 - $6000 (Shoots two bullets at a time)
Any upgrades purchased on weapons will carry through a death, even if you lose the newly picked up weapon in the process.