As soon as you start a new mission, on the Briefing Screen press to choose what weapons you wish to take into the mission. Then go to Silverballer and press to upgrade it. There are around 10-12 upgrades to buy and in the case of the Silverballer you will need $1,525,000 to upgrade the gun fully. Press to buy the upgrades then after you have bought them all, press to accept - if you do not do this then none of the upgrades will be bought. After 'Accepting', press to take that weapon into the level. Complete the level and the trophy will pop during the end-of-mission Statistics Screen.
Note: You can only upgrade a weapon fully until you have completed Mission 8: Till Death do us Part as that's when Tier 4 of the shop opens.
The Roadmap does not recommend earning all of the upgrade related trophies via this method, because it will mean that you will have to complete a level on 4 separate occasions, which will add more time to achieving the . However, this method should only be used if for some reason you are low on cash.
Simply upgrade any weapon to the max, complete the mission and let the trophy. Soon as the trophy pops, do not save your game, exit out of the game and reload your save. You will then have the funds to upgrade another weapon fully. Repeat this process for all 4 weapons in order achieve all of the upgrade related trophies, but only if you can not afford to upgrade all of the weapons in one go.