Weapons can become upgraded when you have completed a level and earned the required amount of Hunter Bucks doing so. You will be heading in the right direction if you follow the game's recommmendations in regards to what it is requiring you to upgrade as it will prepare and enable you to take part in future missions. If you decide however to deviate from the recommended path you will find yourself having to grind away in levels you have already completed to get the required Hunter Bucks.