In order to upgrade/level up weapons, both melee and ranged, you need to play the Operations mode.
First, switch to that mode through the terminal found by the Thunderhawk embarkation point.
Next, go to the Armouring Hall to set up your character. Once there, you can customize your character and select your weapon. The Bulwark and Assault characters are your close melee weapon specialists, but everyone except the Heavy has access to a melee weapon. Select whichever weapon you want on your character, either the Combat Knife, Chainsword, Power Sword, Power Fist, or Thunder Hammer (not all characters have access to all weapons).
Once you've selected your weapon, you'll see two tabs labeled Version & Perks. Version is what we are concerned with. You'll also see 0 (Standard), 1500 (Master Crafted), 7500 (Artificier), and 17000 (Relic). That is the amount of experience needed to upgrade your weapon to that tier. Once you've got your character set up, go and play Operations till you get enough EXP to upgrade. Be sure and use all your weapons during the mission. If you only use, for example, your Primary weapon, but forget to use your Melee or Secondary, you'll only get exp for the Primary.
Once you've gotten 1500 exp with that weapon, go back to the Armouring Hall and back into the Customization. You'll have gotten a Mastery point that can be used to buy a Perk. Now, you'll be able to buy the next tier of weapon with Requisition Points, gained from completing Operations. You can continue to use your Mastered weapon and gain overall experience for the bar, but weapons in higher tiers are just plain better and you should save Requisition Points to buy a new one. Plus, switching weapons allows you to gain more Mastery points (eventually) and that means more Perks and that means you increase in strength and capabilities.
HOWEVER...you'll also need Armory Data to upgrade to the next tier of weapon. To go to Master Crafted, you'll need a green AD. Green ADs can be obtained from Minimal & Average operations, AND through completing the Trials for each character, so you'll be able to get 18 of them without doing Operations. In order to get the Purple & Orange Armoury Datas, you'll need to play the Substantial and Ruthless Operations. Those can either be found within the mission or given as a reward for completing the Operation. You'll need to either find a Servoskull (floating skull with red eyes) or beat a miniboss to collect an Armoury Data. The mission also does need to be completed in order to collect the data. No collecting and then failing the mission.
Weapon exp and masteries are cumulative through all your characters, so you won't have to level them up for each class.
The following is the amount of exp needed to max the bar for each of the Melee Weapons:
25700
Chainsword
Thunder Hammer
Power Fist
Combat Knife
28600
Power Sword